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### Evaluating Expressions Part 1

"Evaluation of an expression" generally means "simplifying an expression down to a final numerical value". In this lesson, we'll concentrate on the "plug" aspect of evaluation: plugging in values for variables, then computing the value of an expression for a given value of the variable. A variable is a letter, for example x, y or z, and it represents an unspecified numeric value. To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations prescribed by the expression. Usually the only difficult part in evaluation is in keeping track of the minus signs. It is strongly recommended to use parenthesis generously, for clarity and accuracy.
We will focus on polynomial functions in one variable, for now.

Let's say:  A girl is 5 years older than her younger sister.
If the age of the girl is "X",  the age of her younger sister will be X - 5.
So if the girl is 15 years old, her younger sister is 15 - 5 = 10 years old.

Formulas are also expressions. For example, the area of a rectangle = L x B where L is the length and B is the width or breadth.
If the rectangle is 10 m long and 5 m in breadth, its area = 10 x 5 square metres = 50 square metres
If the rectangle is 20 m long and 2 m in breadth, its area = 20 x 2 square metres = 40 square metres
Translating phrases into algebraic expressions is a very critical skill.

Being able to use concepts from algebra in day to day life is a fundamental skill and expressions are a big part of it.
Being able to reduce and simplify expressions with adeptness takes patience and practice.

In this set of examples we take a look at simple cases of evaluating polynomial expressions in one variable (x) for given values of the variable 'x'. Evaluating expressions is also critical while plotting the curves for polynomials on graph paper.
One evaluates the value of the expression at various points, for x = -1, -2, 0 , 1, 2, etc (as an example) and then computes a large number of points on the curve, then proceeds to draw it on graph paper.

Here we shouw a few examples of evaluating algebraic expressions. For those who are aware of functions, all of these are polynomials, hence One-to-One functions.
Once you're done with these examples you might also want to take a look at some more examples over here.

### Example 1: Consider the Polynomial: -6 +8x -3x2

Evaluate the expression for x=-15, x=2, x=9, x=-4

Evaluating the given expression for x = -15
Let us compute. The required value = -6 +8*(-15) -3*(-15)2
=-801

Evaluating the given expression for x = 2
Let us compute. The required value = -6 +8*(2) -3*(2)2
=-2

Evaluating the given expression for x = 9
Let us compute. The required value = -6 +8*(9) -3*(9)2
=-177

Evaluating the given expression for x = -4
Let us compute. The required value = -6 +8*(-4) -3*(-4)2
=-86

### Example 2: Consider the Polynomial: - +7x +4x2

Evaluate the expression for x=-12, x=-2, x=2, x=-20, x=1

Evaluating the given expression for x = -12
Let us compute. The required value = (-12)0 +7*(-12) +4*(-12)2
=491

Evaluating the given expression for x = -2
Let us compute. The required value = (-2)0 +7*(-2) +4*(-2)2
=1

Evaluating the given expression for x = 2
Let us compute. The required value = (2)0 +7*(2) +4*(2)2
=29

Evaluating the given expression for x = -20
Let us compute. The required value = (-20)0 +7*(-20) +4*(-20)2
=1459

Evaluating the given expression for x = 1
Let us compute. The required value = (1)0 +7*(1) +4*(1)2
=10

### Example 3: Consider the Polynomial: -10 -x +3x2

Evaluate the expression for x=-2, x=-10, x=6, x=13, x=-16

Evaluating the given expression for x = -2
Let us compute. The required value = -10 (-2) +3*(-2)2
=4

Evaluating the given expression for x = -10
Let us compute. The required value = -10 (-10) +3*(-10)2
=300

Evaluating the given expression for x = 6
Let us compute. The required value = -10 (6) +3*(6)2
=92

Evaluating the given expression for x = 13
Let us compute. The required value = -10 (13) +3*(13)2
=484

Evaluating the given expression for x = -16
Let us compute. The required value = -10 (-16) +3*(-16)2
=774

### Example 4: Consider the Polynomial: -2 +4x -9x2

Evaluate the expression for x=17, x=11, x=-20, x=2

Evaluating the given expression for x = 17
Let us compute. The required value = -2 +4*(17) -9*(17)2
=-2535

Evaluating the given expression for x = 11
Let us compute. The required value = -2 +4*(11) -9*(11)2
=-1047

Evaluating the given expression for x = -20
Let us compute. The required value = -2 +4*(-20) -9*(-20)2
=-3682

Evaluating the given expression for x = 2
Let us compute. The required value = -2 +4*(2) -9*(2)2
=-30

### Example 5: Consider the Polynomial: -2 -5x -3x2

Evaluate the expression for x=-18, x=-14, x=-9, x=14, x=19

Evaluating the given expression for x = -18
Let us compute. The required value = -2 -5*(-18) -3*(-18)2
=-884

Evaluating the given expression for x = -14
Let us compute. The required value = -2 -5*(-14) -3*(-14)2
=-520

Evaluating the given expression for x = -9
Let us compute. The required value = -2 -5*(-9) -3*(-9)2
=-200

Evaluating the given expression for x = 14
Let us compute. The required value = -2 -5*(14) -3*(14)2
=-660

Evaluating the given expression for x = 19
Let us compute. The required value = -2 -5*(19) -3*(19)2
=-1180

### Example 6: Consider the Polynomial: +7 -2x -8x2

Evaluate the expression for x=15, x=-20, x=16, x=1, x=-15

Evaluating the given expression for x = 15
Let us compute. The required value = +7 -2*(15) -8*(15)2
=-1823

Evaluating the given expression for x = -20
Let us compute. The required value = +7 -2*(-20) -8*(-20)2
=-3153

Evaluating the given expression for x = 16
Let us compute. The required value = +7 -2*(16) -8*(16)2
=-2073

Evaluating the given expression for x = 1
Let us compute. The required value = +7 -2*(1) -8*(1)2
=-3

Evaluating the given expression for x = -15
Let us compute. The required value = +7 -2*(-15) -8*(-15)2
=-1763

### Example 7: Consider the Polynomial: +6 -5x

Evaluate the expression for x=-14, x=19, x=-18, x=-4

Evaluating the given expression for x = -14
Let us compute. The required value = +6 -5*(-14)
=76

Evaluating the given expression for x = 19
Let us compute. The required value = +6 -5*(19)
=-89

Evaluating the given expression for x = -18
Let us compute. The required value = +6 -5*(-18)
=96

Evaluating the given expression for x = -4
Let us compute. The required value = +6 -5*(-4)
=26

### Example 8: Consider the Polynomial: -10 +7x +2x2

Evaluate the expression for x=-19, x=-1, x=-6, x=19

Evaluating the given expression for x = -19
Let us compute. The required value = -10 +7*(-19) +2*(-19)2
=579

Evaluating the given expression for x = -1
Let us compute. The required value = -10 +7*(-1) +2*(-1)2
=-15

Evaluating the given expression for x = -6
Let us compute. The required value = -10 +7*(-6) +2*(-6)2
=20

Evaluating the given expression for x = 19
Let us compute. The required value = -10 +7*(19) +2*(19)2
=845

### Example 9: Consider the Polynomial: +4 +6x +9x2

Evaluate the expression for x=-13, x=-7, x=19, x=0, x=-8

Evaluating the given expression for x = -13
Let us compute. The required value = +4 +6*(-13) +9*(-13)2
=1447

Evaluating the given expression for x = -7
Let us compute. The required value = +4 +6*(-7) +9*(-7)2
=403

Evaluating the given expression for x = 19
Let us compute. The required value = +4 +6*(19) +9*(19)2
=3367

Evaluating the given expression for x = 0
Let us compute. The required value = +4 +6*(0) +9*(0)2
=4

Evaluating the given expression for x = -8
Let us compute. The required value = +4 +6*(-8) +9*(-8)2
=532

### Example 10: Consider the Polynomial: +3 +7x +2x2

Evaluate the expression for x=-4, x=16, x=19, x=14

Evaluating the given expression for x = -4
Let us compute. The required value = +3 +7*(-4) +2*(-4)2
=7

Evaluating the given expression for x = 16
Let us compute. The required value = +3 +7*(16) +2*(16)2
=627

Evaluating the given expression for x = 19
Let us compute. The required value = +3 +7*(19) +2*(19)2
=858

Evaluating the given expression for x = 14
Let us compute. The required value = +3 +7*(14) +2*(14)2
=493

### Example 11: Consider the Polynomial: -8 -2x -6x2

Evaluate the expression for x=-9, x=12, x=-12, x=14, x=5

Evaluating the given expression for x = -9
Let us compute. The required value = -8 -2*(-9) -6*(-9)2
=-476

Evaluating the given expression for x = 12
Let us compute. The required value = -8 -2*(12) -6*(12)2
=-896

Evaluating the given expression for x = -12
Let us compute. The required value = -8 -2*(-12) -6*(-12)2
=-848

Evaluating the given expression for x = 14
Let us compute. The required value = -8 -2*(14) -6*(14)2
=-1212

Evaluating the given expression for x = 5
Let us compute. The required value = -8 -2*(5) -6*(5)2
=-168

### Example 12: Consider the Polynomial: +9 +4x +4x2

Evaluate the expression for x=-10, x=1, x=-12, x=-16, x=16

Evaluating the given expression for x = -10
Let us compute. The required value = +9 +4*(-10) +4*(-10)2
=369

Evaluating the given expression for x = 1
Let us compute. The required value = +9 +4*(1) +4*(1)2
=17

Evaluating the given expression for x = -12
Let us compute. The required value = +9 +4*(-12) +4*(-12)2
=537

Evaluating the given expression for x = -16
Let us compute. The required value = +9 +4*(-16) +4*(-16)2
=969

Evaluating the given expression for x = 16
Let us compute. The required value = +9 +4*(16) +4*(16)2
=1097

### Example 13: Consider the Polynomial: +6 +7x +9x2

Evaluate the expression for x=17, x=2, x=-20, x=14, x=-5

Evaluating the given expression for x = 17
Let us compute. The required value = +6 +7*(17) +9*(17)2
=2726

Evaluating the given expression for x = 2
Let us compute. The required value = +6 +7*(2) +9*(2)2
=56

Evaluating the given expression for x = -20
Let us compute. The required value = +6 +7*(-20) +9*(-20)2
=3466

Evaluating the given expression for x = 14
Let us compute. The required value = +6 +7*(14) +9*(14)2
=1868

Evaluating the given expression for x = -5
Let us compute. The required value = +6 +7*(-5) +9*(-5)2
=196

### Example 14: Consider the Polynomial: -9 -9x

Evaluate the expression for x=-15, x=8, x=-11, x=3, x=13

Evaluating the given expression for x = -15
Let us compute. The required value = -9 -9*(-15)
=126

Evaluating the given expression for x = 8
Let us compute. The required value = -9 -9*(8)
=-81

Evaluating the given expression for x = -11
Let us compute. The required value = -9 -9*(-11)
=90

Evaluating the given expression for x = 3
Let us compute. The required value = -9 -9*(3)
=-36

Evaluating the given expression for x = 13
Let us compute. The required value = -9 -9*(13)
=-126

### Example 15: Consider the Polynomial: -8 +2x +2x2

Evaluate the expression for x=1, x=-19, x=-10, x=2, x=-9

Evaluating the given expression for x = 1
Let us compute. The required value = -8 +2*(1) +2*(1)2
=-4

Evaluating the given expression for x = -19
Let us compute. The required value = -8 +2*(-19) +2*(-19)2
=676

Evaluating the given expression for x = -10
Let us compute. The required value = -8 +2*(-10) +2*(-10)2
=172

Evaluating the given expression for x = 2
Let us compute. The required value = -8 +2*(2) +2*(2)2
=4

Evaluating the given expression for x = -9
Let us compute. The required value = -8 +2*(-9) +2*(-9)2
=136

### Example 16: Consider the Polynomial: +6 +3x -2x2

Evaluate the expression for x=17, x=-14, x=-16, x=4

Evaluating the given expression for x = 17
Let us compute. The required value = +6 +3*(17) -2*(17)2
=-521

Evaluating the given expression for x = -14
Let us compute. The required value = +6 +3*(-14) -2*(-14)2
=-428

Evaluating the given expression for x = -16
Let us compute. The required value = +6 +3*(-16) -2*(-16)2
=-554

Evaluating the given expression for x = 4
Let us compute. The required value = +6 +3*(4) -2*(4)2
=-14

### Example 17: Consider the Polynomial: -5 -x -5x2

Evaluate the expression for x=-1, x=0, x=-10, x=4, x=-16

Evaluating the given expression for x = -1
Let us compute. The required value = -5 (-1) -5*(-1)2
=-9

Evaluating the given expression for x = 0
Let us compute. The required value = -5 (0) -5*(0)2
=-5

Evaluating the given expression for x = -10
Let us compute. The required value = -5 (-10) -5*(-10)2
=-495

Evaluating the given expression for x = 4
Let us compute. The required value = -5 (4) -5*(4)2
=-89

Evaluating the given expression for x = -16
Let us compute. The required value = -5 (-16) -5*(-16)2
=-1269

### Example 18: Consider the Polynomial: -9 +6x +3x2

Evaluate the expression for x=-5, x=-1, x=14

Evaluating the given expression for x = -5
Let us compute. The required value = -9 +6*(-5) +3*(-5)2
=36

Evaluating the given expression for x = -1
Let us compute. The required value = -9 +6*(-1) +3*(-1)2
=-12

Evaluating the given expression for x = 14
Let us compute. The required value = -9 +6*(14) +3*(14)2
=663

### Example 19: Consider the Polynomial: -8x -7x2

Evaluate the expression for x=19, x=0, x=-3, x=8, x=-18

Evaluating the given expression for x = 19
Let us compute. The required value = (19)0 -8*(19) -7*(19)2
=-2678

Evaluating the given expression for x = 0
Let us compute. The required value = (0)0 -8*(0) -7*(0)2
=1

Evaluating the given expression for x = -3
Let us compute. The required value = (-3)0 -8*(-3) -7*(-3)2
=-38

Evaluating the given expression for x = 8
Let us compute. The required value = (8)0 -8*(8) -7*(8)2
=-511

Evaluating the given expression for x = -18
Let us compute. The required value = (-18)0 -8*(-18) -7*(-18)2
=-2123

### Example 20: Consider the Polynomial: +6 -6x -6x2

Evaluate the expression for x=17, x=-20, x=-6, x=0, x=-16

Evaluating the given expression for x = 17
Let us compute. The required value = +6 -6*(17) -6*(17)2
=-1830

Evaluating the given expression for x = -20
Let us compute. The required value = +6 -6*(-20) -6*(-20)2
=-2274

Evaluating the given expression for x = -6
Let us compute. The required value = +6 -6*(-6) -6*(-6)2
=-174

Evaluating the given expression for x = 0
Let us compute. The required value = +6 -6*(0) -6*(0)2
=6

Evaluating the given expression for x = -16
Let us compute. The required value = +6 -6*(-16) -6*(-16)2
=-1434

### Example 21: Consider the Polynomial: -3 +5x

Evaluate the expression for x=12, x=10, x=2, x=0, x=-7

Evaluating the given expression for x = 12
Let us compute. The required value = -3 +5*(12)
=57

Evaluating the given expression for x = 10
Let us compute. The required value = -3 +5*(10)
=47

Evaluating the given expression for x = 2
Let us compute. The required value = -3 +5*(2)
=7

Evaluating the given expression for x = 0
Let us compute. The required value = -3 +5*(0)
=-3

Evaluating the given expression for x = -7
Let us compute. The required value = -3 +5*(-7)
=-38

### Example 22: Consider the Polynomial: +6 +5x +3x2

Evaluate the expression for x=12, x=19, x=14, x=18

Evaluating the given expression for x = 12
Let us compute. The required value = +6 +5*(12) +3*(12)2
=498

Evaluating the given expression for x = 19
Let us compute. The required value = +6 +5*(19) +3*(19)2
=1184

Evaluating the given expression for x = 14
Let us compute. The required value = +6 +5*(14) +3*(14)2
=664

Evaluating the given expression for x = 18
Let us compute. The required value = +6 +5*(18) +3*(18)2
=1068

### Example 23: Consider the Polynomial: -6 -10x -5x2

Evaluate the expression for x=-15, x=11, x=-8, x=8, x=-13

Evaluating the given expression for x = -15
Let us compute. The required value = -6 -10*(-15) -5*(-15)2
=-981

Evaluating the given expression for x = 11
Let us compute. The required value = -6 -10*(11) -5*(11)2
=-721

Evaluating the given expression for x = -8
Let us compute. The required value = -6 -10*(-8) -5*(-8)2
=-246

Evaluating the given expression for x = 8
Let us compute. The required value = -6 -10*(8) -5*(8)2
=-406

Evaluating the given expression for x = -13
Let us compute. The required value = -6 -10*(-13) -5*(-13)2
=-721

### Example 24: Consider the Polynomial: +8 +3x +5x2

Evaluate the expression for x=10, x=-18, x=-11, x=15, x=8

Evaluating the given expression for x = 10
Let us compute. The required value = +8 +3*(10) +5*(10)2
=538

Evaluating the given expression for x = -18
Let us compute. The required value = +8 +3*(-18) +5*(-18)2
=1574

Evaluating the given expression for x = -11
Let us compute. The required value = +8 +3*(-11) +5*(-11)2
=580

Evaluating the given expression for x = 15
Let us compute. The required value = +8 +3*(15) +5*(15)2
=1178

Evaluating the given expression for x = 8
Let us compute. The required value = +8 +3*(8) +5*(8)2
=352

### Example 25: Consider the Polynomial: +5 +6x -4x2

Evaluate the expression for x=3, x=19, x=-16, x=18, x=-8

Evaluating the given expression for x = 3
Let us compute. The required value = +5 +6*(3) -4*(3)2
=-13

Evaluating the given expression for x = 19
Let us compute. The required value = +5 +6*(19) -4*(19)2
=-1325

Evaluating the given expression for x = -16
Let us compute. The required value = +5 +6*(-16) -4*(-16)2
=-1115

Evaluating the given expression for x = 18
Let us compute. The required value = +5 +6*(18) -4*(18)2
=-1183

Evaluating the given expression for x = -8
Let us compute. The required value = +5 +6*(-8) -4*(-8)2
=-299