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### Polynomial Expressions: The Remainder Theorem and Factor Theorem

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

The factor theorem states that an algebraic polynomial  has a factor  if and only if   (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.

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### 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

### 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 + 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

### 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