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Polynomial Expressions: The Remainder Theorem and Factor Theorem

The remainder theorem states that the remainder left on dividng a polynomial f(x) by  (x - k)  equals f(k). 

The factor theorem states that an algebraic polynomial {\displaystyle f(x)}f(x) has a factor {\displaystyle (x-k)}(x - k) if and only if {\displaystyle f(k)=0}f(k)=0  (implying that 'k' is a root). 

We will take a look at a few such examples. 
We will also see a few interesting examples which demonstrate how the clever application of this theorem can simply the process of division of polynomials by terms of the form (x-a). 
Instead of attempting to find the quotient first, which can be extremely tricky in the case of long polynomials; we first compute the remainder and then work backwards to find the Quotient. 

Here are some pages with simple examples of evaluating basic algebraic polynomial expressions: evaluating expressions 1 and 2. 


Example 1: Consider the Polynomial: -5x -3

Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= -5*(-8) -3*(-8)0
=37


Let us verifying or check this: -5 (x + 8) + 37 = -5x - 40 + 37 = -5x - 3 = original polynomial. 

So the quotient is -5 and the remainder is -3. An important application of the remainder theorem is that it actually helps us factorize polynomials more easily. 

For example, in this case if we know that the remainder division of 37, we can rewrire -5x - 3 as 37 - 40 - 5x.  = 37 - 5 (x + 8) 
Hence the quotient of division is -5 and the remainder is 37. 

-----

Consider the Polynomial: -5x -3


Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -5*(-2) -3*(-2)0
=7



Find the remainder on dividing the above polynomial by x - 9

Using the remainder theorem, the remainder left = f(9) =

= -5*(9) -3*(9)0
=-48




Example 2: Consider the Polynomial: -4x2 +3x -4


Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= -4*(6)2 +3*(6) -4*(6)0
=-130


Example of how knowledge of this remainder makes divisions over here. 

We rewrite this expression as the following (where Q is the quotient of division) 

 -4x2 +3x -4 = -130 + Q(x-6) => Q (x-6) = -4x2 +3x -4 + 130 =  -4x2 +3x + 126 = -4x (x-6) - 21x + 126 = -4x (x-6) - 21 (x-6) = (-4x - 21)(x-6) 


So the quotient of this division is   (-4x - 21) and the remainder is -130 

------------------------
 

Consider the Polynomial: -4x2 +3x -4


Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -4*(12)2 +3*(12) -4*(12)0
=-544



Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= -4*(19)2 +3*(19) -4*(19)0
=-1391




Example 3: Consider the Polynomial: -2x2 -3x -


Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -2*(10)2 -3*(10) -(10)0
=-231



Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= -2*(-4)2 -3*(-4) -(-4)0
=-21



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -2*(-2)2 -3*(-2) -(-2)0
=-3




Example 4:  (ICSE 2017)


What must be subtraced from 16x3 -  8x2 + 4x + 7 to make it divisible by (2x + 1)  ?


Point to note, is that division of 

(16x3 - 8x2 + 4x + 7) by  (2x + 1)  is the same as division of  (8x3 - 4x2 + 2x + 7/2) by  (x + 1/2)


The remainder of division for the latter  aka (8x3 - 4x2 + 2x + 7/2) by  (x + 1/2) will be: 

8 x (-1/2)3  - 4 (-1/2) 2  + 2 (-1/2) + 7/2  = -1 -1 -1 + 7/2 = 1/2

Remainder of division of 

(16x3 - 8x2 + 4x + 7) by  (2x + 1)  will be 1.


16x3 -  8x2 + 4x + 6 will be divisible by 2x + 1.


The division is visible here: 







Example 5: (ICSE 2016, Mathematics) 

Find 'k' if  2x3 +3x2 -kx + 5  leaves a remainder of 7 when divided by (x-2). 

Solution: f(x) = 2x3 +3x2 -kx + 5

For this to be divisible by (x-2) 

f(2) = 0 => 2 (2)3 + 3 (2)2 - k(2) + 5 = 7 => 16 + 12 -2k + 5 = 7 

=> 2k = 26

=> k = 13 (answer) 

Example 6: Consider the Polynomial: x2 + 2x -8


Is this divisible by (x +4) ?

Here, we will try to use the factor theorem. 


Using the remainder theorem, the remainder left = f(-4) =

=   (-4)2 + 2(-4) -8 =  16 - 8 - 8 = 0

=> Hence the given polynomial is divisible by (x + 4) 
=>  -4 is a root of the quadratic equation  x2 + 2x -8 = 0



Is this divisible by (x - 2) ?

Here, once again we will try to use the factor theorem. 


Using the remainder theorem, the remainder left = f(2) = 

=   (2)2 + 2(2) -8 =  4 + 4 - 8 = 0

=> Hence the given polynomial is divisible by (x - 2) 
=> 2 is a root of the quadratic equation  x2 + 2x -8 = 0


Here we see the factor theorem at work. 

Example 7: Consider the Polynomial: +x3 +3x2 +4x +2


Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= +(17)3 +3*(17)2 +4*(17) +2*(17)0
=5850



Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= +(-20)3 +3*(-20)2 +4*(-20) +2*(-20)0
=-6878



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= +(-16)3 +3*(-16)2 +4*(-16) +2*(-16)0
=-3390




Example 8: Consider the Polynomial: -4x2 +x +


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -4*(-13)2 +(-13) +(-13)0
=-688



Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= -4*(17)2 +(17) +(17)0
=-1138



Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= -4*(-10)2 +(-10) +(-10)0
=-409




Example 9: Consider the Polynomial: -5x2 +4x -


Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= -5*(-9)2 +4*(-9) -(-9)0
=-442



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -5*(2)2 +4*(2) -(2)0
=-13



Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= -5*(-7)2 +4*(-7) -(-7)0
=-274




Example 10: Consider the Polynomial: -2x3 +3x2 -4


Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= -2*(6)3 +3*(6)2 -4*(6)0
=-328



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -2*(-2)3 +3*(-2)2 -4*(-2)0
=24



Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= -2*(-10)3 +3*(-10)2 -4*(-10)0
=2296




Example 11: Consider the Polynomial: -2x2 +x -


Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -2*(1)2 +(1) -(1)0
=-2



Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= -2*(-8)2 +(-8) -(-8)0
=-137



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -2*(12)2 +(12) -(12)0
=-277




Example 12: Consider the Polynomial: +3x3 -3x2 -3x +4


Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= +3*(-15)3 -3*(-15)2 -3*(-15) +4*(-15)0
=-10751



Find the remainder on dividing the above polynomial by x - 15

Using the remainder theorem, the remainder left = f(15) =

= +3*(15)3 -3*(15)2 -3*(15) +4*(15)0
=9409



Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= +3*(11)3 -3*(11)2 -3*(11) +4*(11)0
=3601




Example 13: Consider the Polynomial: +4x3 +4x2 -5x -


Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= +4*(6)3 +4*(6)2 -5*(6) -(6)0
=977



Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= +4*(-9)3 +4*(-9)2 -5*(-9) -(-9)0
=-2548



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +4*(-5)3 +4*(-5)2 -5*(-5) -(-5)0
=-376




Example 14: Consider the Polynomial: -4x3 +2x2 -5x -5


Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= -4*(-8)3 +2*(-8)2 -5*(-8) -5*(-8)0
=2211



Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= -4*(-7)3 +2*(-7)2 -5*(-7) -5*(-7)0
=1500




Example 15: Consider the Polynomial: -x2 +2x -


Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -(12)2 +2*(12) -(12)0
=-121



Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= -(-9)2 +2*(-9) -(-9)0
=-100




Example 16: Consider the Polynomial: +x2 -5x +4


Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= +(-2)2 -5*(-2) +4*(-2)0
=18



Find the remainder on dividing the above polynomial by x + 19

Using the remainder theorem, the remainder left = f(-19) =

= +(-19)2 -5*(-19) +4*(-19)0
=460




Example 17: Consider the Polynomial: +3x3 -4x


Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= +3*(-15)3 -4*(-15)
=-10065



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= +3*(12)3 -4*(12)
=5136



Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= +3*(-3)3 -4*(-3)
=-69




Example 18: Consider the Polynomial: -4x3 +3x2 -4x +4


Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -4*(10)3 +3*(10)2 -4*(10) +4*(10)0
=-3736



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= -4*(-16)3 +3*(-16)2 -4*(-16) +4*(-16)0
=17220



Find the remainder on dividing the above polynomial by x - 18

Using the remainder theorem, the remainder left = f(18) =

= -4*(18)3 +3*(18)2 -4*(18) +4*(18)0
=-22424




Example 19: Consider the Polynomial: -2x2 +x +3


Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= -2*(-20)2 +(-20) +3*(-20)0
=-817



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= -2*(4)2 +(4) +3*(4)0
=-25



Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -2*(3)2 +(3) +3*(3)0
=-12




Example 20: Consider the Polynomial: +2x2 +3x -2


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= +2*(5)2 +3*(5) -2*(5)0
=63



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= +2*(6)2 +3*(6) -2*(6)0
=88



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +2*(-17)2 +3*(-17) -2*(-17)0
=525




Example 21: Consider the Polynomial: +4x3 -4x2 -4x -4


Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= +4*(-10)3 -4*(-10)2 -4*(-10) -4*(-10)0
=-4364



Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= +4*(-15)3 -4*(-15)2 -4*(-15) -4*(-15)0
=-14344



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +4*(-5)3 -4*(-5)2 -4*(-5) -4*(-5)0
=-584




Example 22: Consider the Polynomial: -4x3 -4


Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -4*(2)3 -4*(2)0
=-36



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= -4*(-5)3 -4*(-5)0
=496




Example 23: Consider the Polynomial: +4x3 -5x2 -5x +4


Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +4*(-5)3 -5*(-5)2 -5*(-5) +4*(-5)0
=-596



Find the remainder on dividing the above polynomial by x - 15

Using the remainder theorem, the remainder left = f(15) =

= +4*(15)3 -5*(15)2 -5*(15) +4*(15)0
=12304



Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= +4*(10)3 -5*(10)2 -5*(10) +4*(10)0
=3454




Example 24: Consider the Polynomial: +4x2 -x +2


Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= +4*(10)2 -(10) +2*(10)0
=392



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= +4*(12)2 -(12) +2*(12)0
=566



Find the remainder on dividing the above polynomial by x - 13

Using the remainder theorem, the remainder left = f(13) =

= +4*(13)2 -(13) +2*(13)0
=665




Example 25: Consider the Polynomial: +3x3 +3x2 +x +3


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= +3*(5)3 +3*(5)2 +(5) +3*(5)0
=458



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= +3*(1)3 +3*(1)2 +(1) +3*(1)0
=10



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= +3*(6)3 +3*(6)2 +(6) +3*(6)0
=765




Example 26: Consider the Polynomial: -x3 +4x2 -3x -


Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -(1)3 +4*(1)2 -3*(1) -(1)0
=-1



Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= -(-10)3 +4*(-10)2 -3*(-10) -(-10)0
=1429



Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -(3)3 +4*(3)2 -3*(3) -(3)0
=-1




Example 27: Consider the Polynomial: +2x2 -2x +4


Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= +2*(1)2 -2*(1) +4*(1)0
=4



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +2*(-17)2 -2*(-17) +4*(-17)0
=616



Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= +2*(-10)2 -2*(-10) +4*(-10)0
=224




Example 28: Consider the Polynomial: -5x2 -4x -5


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= -5*(5)2 -4*(5) -5*(5)0
=-150



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -5*(2)2 -4*(2) -5*(2)0
=-33



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -5*(8)2 -4*(8) -5*(8)0
=-357




Example 29: Consider the Polynomial: +4x3 +2x2 -3x -2


Find the remainder on dividing the above polynomial by x + 14

Using the remainder theorem, the remainder left = f(-14) =

= +4*(-14)3 +2*(-14)2 -3*(-14) -2*(-14)0
=-10544



Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= +4*(-8)3 +2*(-8)2 -3*(-8) -2*(-8)0
=-1898




Example 30: Consider the Polynomial: -2x3 +x2 -x +2


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -2*(-13)3 +(-13)2 -(-13) +2*(-13)0
=4578



Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -2*(3)3 +(3)2 -(3) +2*(3)0
=-46



Find the remainder on dividing the above polynomial by x - 16

Using the remainder theorem, the remainder left = f(16) =

= -2*(16)3 +(16)2 -(16) +2*(16)0
=-7950




Example 31: Consider the Polynomial: -x2 +4x +2


Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -(-12)2 +4*(-12) +2*(-12)0
=-190



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= -(4)2 +4*(4) +2*(4)0
=2




Example 32: Consider the Polynomial: +3x2 -4x +4


Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= +3*(19)2 -4*(19) +4*(19)0
=1011



Find the remainder on dividing the above polynomial by x - 14

Using the remainder theorem, the remainder left = f(14) =

= +3*(14)2 -4*(14) +4*(14)0
=536




Example 33: Consider the Polynomial: -3x3 +3x2 +4x -5


Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= -3*(19)3 +3*(19)2 +4*(19) -5*(19)0
=-19423



Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= -3*(-7)3 +3*(-7)2 +4*(-7) -5*(-7)0
=1143



Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -3*(-12)3 +3*(-12)2 +4*(-12) -5*(-12)0
=5563




Example 34: Consider the Polynomial: +x3 +4x2 -2x


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= +(-13)3 +4*(-13)2 -2*(-13)
=-1495



Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= +(-12)3 +4*(-12)2 -2*(-12)
=-1128



Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= +(-9)3 +4*(-9)2 -2*(-9)
=-387




Example 35: Consider the Polynomial: +2x3 +4x2 -4x +4


Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= +2*(-20)3 +4*(-20)2 -4*(-20) +4*(-20)0
=-14316



Find the remainder on dividing the above polynomial by x - 14

Using the remainder theorem, the remainder left = f(14) =

= +2*(14)3 +4*(14)2 -4*(14) +4*(14)0
=6220



Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= +2*(-11)3 +4*(-11)2 -4*(-11) +4*(-11)0
=-2130




Example 36: Consider the Polynomial: -3x2 +4x +3


Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -3*(-12)2 +4*(-12) +3*(-12)0
=-477



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= -3*(-17)2 +4*(-17) +3*(-17)0
=-932



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= -3*(6)2 +4*(6) +3*(6)0
=-81




Example 37: Consider the Polynomial: -4x3 -x2 -2x +2


Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= -4*(-15)3 -(-15)2 -2*(-15) +2*(-15)0
=13307



Find the remainder on dividing the above polynomial by x - 9

Using the remainder theorem, the remainder left = f(9) =

= -4*(9)3 -(9)2 -2*(9) +2*(9)0
=-3013



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -4*(-2)3 -(-2)2 -2*(-2) +2*(-2)0
=34




Example 38: Consider the Polynomial: +3x3 -3x2 -2x -4


Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= +3*(10)3 -3*(10)2 -2*(10) -4*(10)0
=2676



Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= +3*(-13)3 -3*(-13)2 -2*(-13) -4*(-13)0
=-7076



Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= +3*(-8)3 -3*(-8)2 -2*(-8) -4*(-8)0
=-1716




Example 39: Consider the Polynomial: +2x2 +4x


Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= +2*(-12)2 +4*(-12)
=240



Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +2*(-4)2 +4*(-4)
=16



Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= +2*(-6)2 +4*(-6)
=48




Example 40: Consider the Polynomial: +4x2 -3x -5


Find the remainder on dividing the above polynomial by x - 13

Using the remainder theorem, the remainder left = f(13) =

= +4*(13)2 -3*(13) -5*(13)0
=632



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= +4*(4)2 -3*(4) -5*(4)0
=47



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +4*(-17)2 -3*(-17) -5*(-17)0
=1202




Example 41: Consider the Polynomial: -5x3 -5x2 -2x


Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -5*(8)3 -5*(8)2 -2*(8)
=-2896



Find the remainder on dividing the above polynomial by x + 19

Using the remainder theorem, the remainder left = f(-19) =

= -5*(-19)3 -5*(-19)2 -2*(-19)
=32528



Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -5*(3)3 -5*(3)2 -2*(3)
=-186




Example 42: Consider the Polynomial: -5x2 -5x -3


Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= -5*(-15)2 -5*(-15) -3*(-15)0
=-1053



Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= -5*(19)2 -5*(19) -3*(19)0
=-1903



Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -5*(-13)2 -5*(-13) -3*(-13)0
=-783




Example 43: Consider the Polynomial: +x +2


Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= +(-10) +2*(-10)0
=-8



Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= +(17) +2*(17)0
=19



Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= +(-15) +2*(-15)0
=-13




Example 44: Consider the Polynomial: +x2 +3x +2


Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= +(-11)2 +3*(-11) +2*(-11)0
=90



Find the remainder on dividing the above polynomial by x - 16

Using the remainder theorem, the remainder left = f(16) =

= +(16)2 +3*(16) +2*(16)0
=306



Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= +(-9)2 +3*(-9) +2*(-9)0
=56




Example 45: Consider the Polynomial: -3x2 -5x -5


Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -3*(1)2 -5*(1) -5*(1)0
=-13



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= -3*(-17)2 -5*(-17) -5*(-17)0
=-787



Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -3*(-13)2 -5*(-13) -5*(-13)0
=-447




Example 46: Consider the Polynomial: -5x3 -4x2 -3x -


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= -5*(5)3 -4*(5)2 -3*(5) -(5)0
=-741



Find the remainder on dividing the above polynomial by x + 19

Using the remainder theorem, the remainder left = f(-19) =

= -5*(-19)3 -4*(-19)2 -3*(-19) -(-19)0
=32907



Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= -5*(-8)3 -4*(-8)2 -3*(-8) -(-8)0
=2327




Example 47: Consider the Polynomial: -x2 +3x


Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= -(-11)2 +3*(-11)
=-154



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= -(6)2 +3*(6)
=-18



Find the remainder on dividing the above polynomial by x + 1

Using the remainder theorem, the remainder left = f(-1) =

= -(-1)2 +3*(-1)
=-4




Example 48: Consider the Polynomial: +4x2 +2x -3


Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +4*(-5)2 +2*(-5) -3*(-5)0
=87



Find the remainder on dividing the above polynomial by x - 18

Using the remainder theorem, the remainder left = f(18) =

= +4*(18)2 +2*(18) -3*(18)0
=1329



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= +4*(2)2 +2*(2) -3*(2)0
=17




Example 49: Consider the Polynomial: +2x2 -2x -2


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +2*(-4)2 -2*(-4) -2*(-4)0
=38



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= +2*(6)2 -2*(6) -2*(6)0
=58



Find the remainder on dividing the above polynomial by x - 13

Using the remainder theorem, the remainder left = f(13) =

= +2*(13)2 -2*(13) -2*(13)0
=310




Example 50: Consider the Polynomial: -x3 -4x2 -x +3


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -(-13)3 -4*(-13)2 -(-13) +3*(-13)0
=1537



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -(-2)3 -4*(-2)2 -(-2) +3*(-2)0
=-3



Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= -(-3)3 -4*(-3)2 -(-3) +3*(-3)0
=-3




Example 51: Consider the Polynomial: +2x2 +2x +4


Find the remainder on dividing the above polynomial by x + 1

Using the remainder theorem, the remainder left = f(-1) =

= +2*(-1)2 +2*(-1) +4*(-1)0
=4



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= +2*(2)2 +2*(2) +4*(2)0
=16



Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= +2*(-20)2 +2*(-20) +4*(-20)0
=764




Example 52: Consider the Polynomial: +3x3 -2x2 -x +3


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= +3*(-13)3 -2*(-13)2 -(-13) +3*(-13)0
=-6913



Find the remainder on dividing the above polynomial by x - 18

Using the remainder theorem, the remainder left = f(18) =

= +3*(18)3 -2*(18)2 -(18) +3*(18)0
=16833



Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= +3*(-3)3 -2*(-3)2 -(-3) +3*(-3)0
=-93




Example 53: Consider the Polynomial: +4x3 +x2 -3x -2


Find the remainder on dividing the above polynomial by x - 15

Using the remainder theorem, the remainder left = f(15) =

= +4*(15)3 +(15)2 -3*(15) -2*(15)0
=13678



Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +4*(-4)3 +(-4)2 -3*(-4) -2*(-4)0
=-230




Example 54: Consider the Polynomial: -4x2 +2x -4


Find the remainder on dividing the above polynomial by x - 13

Using the remainder theorem, the remainder left = f(13) =

= -4*(13)2 +2*(13) -4*(13)0
=-654



Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= -4*(-20)2 +2*(-20) -4*(-20)0
=-1644



Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -4*(-12)2 +2*(-12) -4*(-12)0
=-604




Example 55: Consider the Polynomial: -x2 +3x +3


Find the remainder on dividing the above polynomial by x - 14

Using the remainder theorem, the remainder left = f(14) =

= -(14)2 +3*(14) +3*(14)0
=-151



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -(12)2 +3*(12) +3*(12)0
=-105



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -(8)2 +3*(8) +3*(8)0
=-37




Example 56: Consider the Polynomial: -2x2 -2x -5


Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= -2*(-5)2 -2*(-5) -5*(-5)0
=-45



Find the remainder on dividing the above polynomial by x + 14

Using the remainder theorem, the remainder left = f(-14) =

= -2*(-14)2 -2*(-14) -5*(-14)0
=-369



Find the remainder on dividing the above polynomial by x - 15

Using the remainder theorem, the remainder left = f(15) =

= -2*(15)2 -2*(15) -5*(15)0
=-485




Example 57: Consider the Polynomial: +x3 +x2 +x -3


Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= +(-8)3 +(-8)2 +(-8) -3*(-8)0
=-459



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= +(-2)3 +(-2)2 +(-2) -3*(-2)0
=-9



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= +(8)3 +(8)2 +(8) -3*(8)0
=581




Example 58: Consider the Polynomial: +4x2 +2x -4


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +4*(-4)2 +2*(-4) -4*(-4)0
=52



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +4*(-17)2 +2*(-17) -4*(-17)0
=1118



Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= +4*(-15)2 +2*(-15) -4*(-15)0
=866




Example 59: Consider the Polynomial: +4x +3


Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= +4*(11) +3*(11)0
=47



Find the remainder on dividing the above polynomial by x + 18

Using the remainder theorem, the remainder left = f(-18) =

= +4*(-18) +3*(-18)0
=-69



Find the remainder on dividing the above polynomial by x + 1

Using the remainder theorem, the remainder left = f(-1) =

= +4*(-1) +3*(-1)0
=-1




Example 60: Consider the Polynomial: -5x3 -4x2 -3x +


Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= -5*(-11)3 -4*(-11)2 -3*(-11) +(-11)0
=6205



Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= -5*(-20)3 -4*(-20)2 -3*(-20) +(-20)0
=38461



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -5*(2)3 -4*(2)2 -3*(2) +(2)0
=-61




Example 61: Consider the Polynomial: +3x2 +2x -5


Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= +3*(2)2 +2*(2) -5*(2)0
=11



Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= +3*(-3)2 +2*(-3) -5*(-3)0
=16



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= +3*(-2)2 +2*(-2) -5*(-2)0
=3




Example 62: Consider the Polynomial: -x2 -3x -


Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -(8)2 -3*(8) -(8)0
=-89



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= -(4)2 -3*(4) -(4)0
=-29



Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -(-13)2 -3*(-13) -(-13)0
=-131




Example 63: Consider the Polynomial: +4x2 -2x -5


Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= +4*(-13)2 -2*(-13) -5*(-13)0
=697



Find the remainder on dividing the above polynomial by x - 14

Using the remainder theorem, the remainder left = f(14) =

= +4*(14)2 -2*(14) -5*(14)0
=751



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= +4*(-16)2 -2*(-16) -5*(-16)0
=1051




Example 64: Consider the Polynomial: -2


Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -2*(12)0
=-2



Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= -2*(11)0
=-2



Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= -2*(-6)0
=-2




Example 65: Consider the Polynomial: -5x2 -4x -4


Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= -5*(11)2 -4*(11) -4*(11)0
=-653



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -5*(2)2 -4*(2) -4*(2)0
=-32



Find the remainder on dividing the above polynomial by x + 18

Using the remainder theorem, the remainder left = f(-18) =

= -5*(-18)2 -4*(-18) -4*(-18)0
=-1552




Example 66: Consider the Polynomial: -x3 -4x -2


Find the remainder on dividing the above polynomial by x - 9

Using the remainder theorem, the remainder left = f(9) =

= -(9)3 -4*(9) -2*(9)0
=-767



Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= -(11)3 -4*(11) -2*(11)0
=-1377



Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= -(-11)3 -4*(-11) -2*(-11)0
=1373




Example 67: Consider the Polynomial: +3x3 +3x2 -2x -


Find the remainder on dividing the above polynomial by x - 9

Using the remainder theorem, the remainder left = f(9) =

= +3*(9)3 +3*(9)2 -2*(9) -(9)0
=2411



Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +3*(-4)3 +3*(-4)2 -2*(-4) -(-4)0
=-137



Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= +3*(5)3 +3*(5)2 -2*(5) -(5)0
=439




Example 68: Consider the Polynomial: +x2 -5x -


Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= +(11)2 -5*(11) -(11)0
=65



Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= +(-6)2 -5*(-6) -(-6)0
=65



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= +(4)2 -5*(4) -(4)0
=-5




Example 69: Consider the Polynomial: -4x2 -3x -5


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= -4*(-4)2 -3*(-4) -5*(-4)0
=-57



Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -4*(-12)2 -3*(-12) -5*(-12)0
=-545



Find the remainder on dividing the above polynomial by x + 18

Using the remainder theorem, the remainder left = f(-18) =

= -4*(-18)2 -3*(-18) -5*(-18)0
=-1247




Example 70: Consider the Polynomial: +3x2 -5x


Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +3*(-5)2 -5*(-5)
=100



Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= +3*(3)2 -5*(3)
=12



Find the remainder on dividing the above polynomial by x - 18

Using the remainder theorem, the remainder left = f(18) =

= +3*(18)2 -5*(18)
=882




Example 71: Consider the Polynomial: +3x3 -x2 +4x +3


Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +3*(-17)3 -(-17)2 +4*(-17) +3*(-17)0
=-15093



Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= +3*(10)3 -(10)2 +4*(10) +3*(10)0
=2943



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= +3*(1)3 -(1)2 +4*(1) +3*(1)0
=9




Example 72: Consider the Polynomial: -5x3 -4x2 -x +2


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= -5*(-4)3 -4*(-4)2 -(-4) +2*(-4)0
=262



Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -5*(2)3 -4*(2)2 -(2) +2*(2)0
=-56



Find the remainder on dividing the above polynomial by x - 18

Using the remainder theorem, the remainder left = f(18) =

= -5*(18)3 -4*(18)2 -(18) +2*(18)0
=-30472




Example 73: Consider the Polynomial: -2x3 -3x2 -4


Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= -2*(-20)3 -3*(-20)2 -4*(-20)0
=14796



Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -2*(10)3 -3*(10)2 -4*(10)0
=-2304




Example 74: Consider the Polynomial: +x2 +3x +


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +(-4)2 +3*(-4) +(-4)0
=5



Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= +(17)2 +3*(17) +(17)0
=341



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= +(4)2 +3*(4) +(4)0
=29




Example 75: Consider the Polynomial: -x3 +x2 -4x +4


Find the remainder on dividing the above polynomial by x - 16

Using the remainder theorem, the remainder left = f(16) =

= -(16)3 +(16)2 -4*(16) +4*(16)0
=-3900



Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= -(17)3 +(17)2 -4*(17) +4*(17)0
=-4688



Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= -(-3)3 +(-3)2 -4*(-3) +4*(-3)0
=52




Example 76: Consider the Polynomial: -2x2 -4x -3


Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= -2*(11)2 -4*(11) -3*(11)0
=-289



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -2*(8)2 -4*(8) -3*(8)0
=-163



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= -2*(-5)2 -4*(-5) -3*(-5)0
=-33




Example 77: Consider the Polynomial: -2x3 +x2 -x -2


Find the remainder on dividing the above polynomial by x - 2

Using the remainder theorem, the remainder left = f(2) =

= -2*(2)3 +(2)2 -(2) -2*(2)0
=-16



Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -2*(10)3 +(10)2 -(10) -2*(10)0
=-1912



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= -2*(-16)3 +(-16)2 -(-16) -2*(-16)0
=8462




Example 78: Consider the Polynomial: -3x2 -2x -2


Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -3*(3)2 -2*(3) -2*(3)0
=-35



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -3*(1)2 -2*(1) -2*(1)0
=-7



Find the remainder on dividing the above polynomial by x - 6

Using the remainder theorem, the remainder left = f(6) =

= -3*(6)2 -2*(6) -2*(6)0
=-122




Example 79: Consider the Polynomial: +2x2 +4x -2


Find the remainder on dividing the above polynomial by x + 11

Using the remainder theorem, the remainder left = f(-11) =

= +2*(-11)2 +4*(-11) -2*(-11)0
=196



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= +2*(12)2 +4*(12) -2*(12)0
=334



Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= +2*(-7)2 +4*(-7) -2*(-7)0
=68




Example 80: Consider the Polynomial: +x2 -2x +3


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +(-4)2 -2*(-4) +3*(-4)0
=27



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +(-5)2 -2*(-5) +3*(-5)0
=38



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= +(-16)2 -2*(-16) +3*(-16)0
=291




Example 81: Consider the Polynomial: -5x3 -5x2 -x +2


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= -5*(5)3 -5*(5)2 -(5) +2*(5)0
=-753



Find the remainder on dividing the above polynomial by x + 19

Using the remainder theorem, the remainder left = f(-19) =

= -5*(-19)3 -5*(-19)2 -(-19) +2*(-19)0
=32511




Example 82: Consider the Polynomial: -3x3 +x2 +x -5


Find the remainder on dividing the above polynomial by x - 3

Using the remainder theorem, the remainder left = f(3) =

= -3*(3)3 +(3)2 +(3) -5*(3)0
=-74



Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= -3*(-7)3 +(-7)2 +(-7) -5*(-7)0
=1066



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= -3*(4)3 +(4)2 +(4) -5*(4)0
=-177




Example 83: Consider the Polynomial: -3x2 +x -


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= -3*(-4)2 +(-4) -(-4)0
=-53



Find the remainder on dividing the above polynomial by x - 12

Using the remainder theorem, the remainder left = f(12) =

= -3*(12)2 +(12) -(12)0
=-421



Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= -3*(-6)2 +(-6) -(-6)0
=-115




Example 84: Consider the Polynomial: -5x3 -2x2 -


Find the remainder on dividing the above polynomial by x + 12

Using the remainder theorem, the remainder left = f(-12) =

= -5*(-12)3 -2*(-12)2 -(-12)0
=8351



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= -5*(-17)3 -2*(-17)2 -(-17)0
=23986



Find the remainder on dividing the above polynomial by x + 9

Using the remainder theorem, the remainder left = f(-9) =

= -5*(-9)3 -2*(-9)2 -(-9)0
=3482




Example 85: Consider the Polynomial: +3x3 +3x2 -2x +


Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= +3*(-6)3 +3*(-6)2 -2*(-6) +(-6)0
=-527



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= +3*(-16)3 +3*(-16)2 -2*(-16) +(-16)0
=-11487



Find the remainder on dividing the above polynomial by x + 1

Using the remainder theorem, the remainder left = f(-1) =

= +3*(-1)3 +3*(-1)2 -2*(-1) +(-1)0
=3




Example 86: Consider the Polynomial: -4x3 -x2 +2


Find the remainder on dividing the above polynomial by x + 3

Using the remainder theorem, the remainder left = f(-3) =

= -4*(-3)3 -(-3)2 +2*(-3)0
=101



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -4*(1)3 -(1)2 +2*(1)0
=-3



Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= -4*(-4)3 -(-4)2 +2*(-4)0
=242




Example 87: Consider the Polynomial: -5x2 -3x -5


Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= -5*(-5)2 -3*(-5) -5*(-5)0
=-115



Find the remainder on dividing the above polynomial by x + 19

Using the remainder theorem, the remainder left = f(-19) =

= -5*(-19)2 -3*(-19) -5*(-19)0
=-1753



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -5*(1)2 -3*(1) -5*(1)0
=-13




Example 88: Consider the Polynomial: +2x3 -3x2


Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= +2*(8)3 -3*(8)2
=832



Find the remainder on dividing the above polynomial by x + 16

Using the remainder theorem, the remainder left = f(-16) =

= +2*(-16)3 -3*(-16)2
=-8960



Find the remainder on dividing the above polynomial by x + 6

Using the remainder theorem, the remainder left = f(-6) =

= +2*(-6)3 -3*(-6)2
=-540




Example 89: Consider the Polynomial: -x3 -x -


Find the remainder on dividing the above polynomial by x - 16

Using the remainder theorem, the remainder left = f(16) =

= -(16)3 -(16) -(16)0
=-4113



Find the remainder on dividing the above polynomial by x - 4

Using the remainder theorem, the remainder left = f(4) =

= -(4)3 -(4) -(4)0
=-69



Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= -(19)3 -(19) -(19)0
=-6879




Example 90: Consider the Polynomial: +3x2 -5x +2


Find the remainder on dividing the above polynomial by x + 4

Using the remainder theorem, the remainder left = f(-4) =

= +3*(-4)2 -5*(-4) +2*(-4)0
=70



Find the remainder on dividing the above polynomial by x + 20

Using the remainder theorem, the remainder left = f(-20) =

= +3*(-20)2 -5*(-20) +2*(-20)0
=1302



Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= +3*(17)2 -5*(17) +2*(17)0
=784




Example 91: Consider the Polynomial: -2x2 +4x -3


Find the remainder on dividing the above polynomial by x + 7

Using the remainder theorem, the remainder left = f(-7) =

= -2*(-7)2 +4*(-7) -3*(-7)0
=-129



Find the remainder on dividing the above polynomial by x - 16

Using the remainder theorem, the remainder left = f(16) =

= -2*(16)2 +4*(16) -3*(16)0
=-451



Find the remainder on dividing the above polynomial by x - 1

Using the remainder theorem, the remainder left = f(1) =

= -2*(1)2 +4*(1) -3*(1)0
=-1




Example 92: Consider the Polynomial: +2x3 -x2 -x +3


Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= +2*(5)3 -(5)2 -(5) +3*(5)0
=223



Find the remainder on dividing the above polynomial by x + 5

Using the remainder theorem, the remainder left = f(-5) =

= +2*(-5)3 -(-5)2 -(-5) +3*(-5)0
=-267



Find the remainder on dividing the above polynomial by x - 11

Using the remainder theorem, the remainder left = f(11) =

= +2*(11)3 -(11)2 -(11) +3*(11)0
=2533




Example 93: Consider the Polynomial: -x2 -4x +3


Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= -(17)2 -4*(17) +3*(17)0
=-354



Find the remainder on dividing the above polynomial by x - 7

Using the remainder theorem, the remainder left = f(7) =

= -(7)2 -4*(7) +3*(7)0
=-74




Example 94: Consider the Polynomial: -4x2 +x


Find the remainder on dividing the above polynomial by x + 8

Using the remainder theorem, the remainder left = f(-8) =

= -4*(-8)2 +(-8)
=-264



Find the remainder on dividing the above polynomial by x + 15

Using the remainder theorem, the remainder left = f(-15) =

= -4*(-15)2 +(-15)
=-915



Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -4*(10)2 +(10)
=-390




Example 95: Consider the Polynomial: +x3 +3x2 -3x +4


Find the remainder on dividing the above polynomial by x - 7

Using the remainder theorem, the remainder left = f(7) =

= +(7)3 +3*(7)2 -3*(7) +4*(7)0
=473



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= +(-17)3 +3*(-17)2 -3*(-17) +4*(-17)0
=-3991



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= +(8)3 +3*(8)2 -3*(8) +4*(8)0
=684




Example 96: Consider the Polynomial: +2x2 -4x -3


Find the remainder on dividing the above polynomial by x - 9

Using the remainder theorem, the remainder left = f(9) =

= +2*(9)2 -4*(9) -3*(9)0
=123



Find the remainder on dividing the above polynomial by x - 5

Using the remainder theorem, the remainder left = f(5) =

= +2*(5)2 -4*(5) -3*(5)0
=27




Example 97: Consider the Polynomial: -3x2 +x -2


Find the remainder on dividing the above polynomial by x - 15

Using the remainder theorem, the remainder left = f(15) =

= -3*(15)2 +(15) -2*(15)0
=-662



Find the remainder on dividing the above polynomial by x + 13

Using the remainder theorem, the remainder left = f(-13) =

= -3*(-13)2 +(-13) -2*(-13)0
=-522



Find the remainder on dividing the above polynomial by x + 10

Using the remainder theorem, the remainder left = f(-10) =

= -3*(-10)2 +(-10) -2*(-10)0
=-312




Example 98: Consider the Polynomial: -5x2 -5x +


Find the remainder on dividing the above polynomial by x - 10

Using the remainder theorem, the remainder left = f(10) =

= -5*(10)2 -5*(10) +(10)0
=-549



Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= -5*(8)2 -5*(8) +(8)0
=-359



Find the remainder on dividing the above polynomial by x + 17

Using the remainder theorem, the remainder left = f(-17) =

= -5*(-17)2 -5*(-17) +(-17)0
=-1359




Example 99: Consider the Polynomial: -2x3 +x2 -3x +4


Find the remainder on dividing the above polynomial by x - 17

Using the remainder theorem, the remainder left = f(17) =

= -2*(17)3 +(17)2 -3*(17) +4*(17)0
=-9584



Find the remainder on dividing the above polynomial by x - 19

Using the remainder theorem, the remainder left = f(19) =

= -2*(19)3 +(19)2 -3*(19) +4*(19)0
=-13410



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= -2*(-2)3 +(-2)2 -3*(-2) +4*(-2)0
=30




Example 100: Consider the Polynomial: +4x2 -5x


Find the remainder on dividing the above polynomial by x - 8

Using the remainder theorem, the remainder left = f(8) =

= +4*(8)2 -5*(8)
=216



Find the remainder on dividing the above polynomial by x + 2

Using the remainder theorem, the remainder left = f(-2) =

= +4*(-2)2 -5*(-2)
=26