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10.3: Evaluate, Simplify, and Translate Expressions (Part 1)


Skills to Develop

  • Evaluate algebraic expressions
  • Identify terms, coefficients, and like terms
  • Simplify expressions by combining like terms
  • Translate word phrases to algebraic expressions

Be prepared!

Before you get started, take this readiness quiz.

  1. Is (n ÷ 5) an expression or an equation? If you missed this problem, review Example 2.1.4.
  2. Simplify (4^5). If you missed this problem, review Example 2.1.6.
  3. Simplify (1 + 8 • 9). If you missed this problem, review Example 2.1.8.

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example (PageIndex{1}): evaluate

Evaluate (x + 7) when

  1. (x = 3)
  2. (x = 12)

Solution

  1. To evaluate, substitute (3) for (x) in the expression, and then simplify.
(x + 7)
Substitute.( extcolor{red}{3} + 7)
Add.(10)

When (x = 3), the expression (x + 7) has a value of (10).

  1. To evaluate, substitute (12) for (x) in the expression, and then simplify.
(x + 7)
Substitute.( extcolor{red}{12} + 7)
Add.(19)

When (x = 12), the expression (x + 7) has a value of (19). Notice that we got different results for parts (a) and (b) even though we started with the same expression. This is because the values used for (x) were different. When we evaluate an expression, the value varies depending on the value used for the variable.

exercise (PageIndex{1})

Evaluate: (y + 4) when

  1. (y = 6)
  2. (y = 15)
Answer a

(10)

Answer b

(19)

exercise (PageIndex{2})

Evaluate: (a − 5) when

  1. (a = 9)
  2. (a = 17)
Answer a

(4)

Answer b

(12)

Example (PageIndex{2})

Evaluate (9x − 2), when

  1. (x = 5)
  2. (x = 1)

Solution

Remember (ab) means (a) times (b), so (9x) means (9) times (x).

  1. To evaluate the expression when (x = 5), we substitute (5) for (x), and then simplify.
(9x - 2)
Substitute ( extcolor{red}{5}) for x.(9 cdot extcolor{red}{5} - 2)
Multiply.(45 - 2)
Subtract.(43)
  1. To evaluate the expression when (x = 1), we substitute (1) for (x), and then simplify.
(9x - 2)
Substitute ( extcolor{red}{1}) for x.(9 cdot extcolor{red}{1} - 2)
Multiply.(9 - 2)
Subtract.(7)

Notice that in part (a) that we wrote (9 • 5) and in part (b) we wrote (9(1)). Both the dot and the parentheses tell us to multiply.

exercise (PageIndex{3})

Evaluate: (8x − 3), when

  1. (x = 2)
  2. (x = 1)
Answer a

(13)

Answer b

(5)

exercise (PageIndex{4})

Evaluate: (4y − 4), when

  1. (y = 3)
  2. (y = 5)
Answer a

(8)

Answer b

(16)

Example (PageIndex{3}): evaluate

Evaluate (x^2) when (x = 10).

Solution

We substitute (10) for (x), and then simplify the expression.

(x^{2})
Substitute ( extcolor{red}{10}) for x.( extcolor{red}{10}^{2})
Use the definition of exponent.(10 cdot 10)
Multiply(100)

When (x = 10), the expression (x^2) has a value of (100).

exercise (PageIndex{5})

Evaluate: (x^2) when (x = 8).

Answer

(64)

exercise (PageIndex{6})

Evaluate: (x^3) when (x = 6).

Answer

(216)

Example (PageIndex{4}): evaluate

Evaluate (2^x) when (x = 5).

Solution

In this expression, the variable is an exponent.

(2^{x})
Substitute ( extcolor{red}{5}) for x.(2^{ extcolor{red}{5}})
Use the definition of exponent.(2 cdot 2 cdot 2 cdot 2 cdot 2)
Multiply(32)

When (x = 5), the expression (2^x) has a value of (32).

exercise (PageIndex{7})

Evaluate: (2^x) when (x = 6).

Answer

(64)

exercise (PageIndex{8})

Evaluate: (3^x) when (x = 4).

Answer

(81)

Example (PageIndex{5}): evaluate

Evaluate (3x + 4y − 6) when (x = 10) and (y = 2).

Solution

This expression contains two variables, so we must make two substitutions.

(3x + 4y − 6)
Substitute ( extcolor{red}{10}) for x and ( extcolor{blue}{2}) for y.(3( extcolor{red}{10}) + 4( extcolor{blue}{2}) − 6)
Multiply.(30 + 8 - 6)
Add and subtract left to right.(32)

When (x = 10) and (y = 2), the expression (3x + 4y − 6) has a value of (32).

exercise (PageIndex{9})

Evaluate: (2x + 5y − 4) when (x = 11) and (y = 3)

Answer

(33)

exercise (PageIndex{10})

Evaluate: (5x − 2y − 9) when (x = 7) and (y = 8)

Answer

(10)

Example (PageIndex{6}): evaluate

Evaluate (2x^2 + 3x + 8) when (x = 4).

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, (2x^2) means (2 • x • x) and is different from the expression ((2x)^2), which means (2x • 2x).

(2x^{2} + 3x + 8)
Substitute ( extcolor{red}{4}) for each x.(2( extcolor{red}{4})^{2} + 3( extcolor{red}{4}) + 8)
Simplify 42.(2(16) + 3(4) + 8)
Multiply.(32 + 12 + 8)
Add.(52)

exercise (PageIndex{11})

Evaluate: (3x^2 + 4x + 1) when (x = 3).

Answer

(40)

exercise (PageIndex{12})

Evaluate: (6x^2 − 4x − 7) when (x = 2).

Answer

(9)

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are (7), (y), (5x^2), (9a), and (13xy).

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term (3x) is (3). When we write (x), the coefficient is (1), since (x = 1 • x). Table (PageIndex{1}) gives the coefficients for each of the terms in the left column.

Table (PageIndex{1})
TermCoefficient
77
9a9
y1
5x25

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table (PageIndex{2}) gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Table (PageIndex{2})
ExpressionTerms
77
yy
x + 7x, 7
2x + 7y + 42x, 7y, 4
3x2 + 4x2 + 5y + 33x2, 4x2, 5y, 3

Example (PageIndex{7}):

Identify each term in the expression (9b + 15x^2 + a + 6). Then identify the coefficient of each term.

Solution

The expression has four terms. They are (9b), (15x^2), (a), and (6).

The coefficient of (9b) is (9).

The coefficient of (15x^2) is (15).

Remember that if no number is written before a variable, the coefficient is (1). So the coefficient of a is (1).

The coefficient of a constant is the constant, so the coefficient of (6) is (6).

exercise (PageIndex{13})

Identify all terms in the given expression, and their coefficients: (4x + 3b + 2)

Answer

The terms are (4x, 3b,) and (2). The coefficients are (4, 3,) and (2).

exercise (PageIndex{14})

Identify all terms in the given expression, and their coefficients: (9a + 13a^2 + a^3)

Answer

The terms are (9a, 13a^2,) and (a^3), The coefficients are (9, 13,) and (1).

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

(5x, 7, n^{2}, 4, 3x, 9n^{2})

Which of these terms are like terms?

  • The terms (7) and (4) are both constant terms.
  • The terms (5x) and (3x) are both terms with (x).
  • The terms (n^2) and (9n^2) both have (n^2).

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms (5x, 7, n^2, 4, 3x, 9n^2, 7) and (4) are like terms, (5x) and (3x) are like terms, and (n^2) and (9n^2) are like terms.

Definition: Like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Example (PageIndex{8}): identify

Identify the like terms:

  1. (y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2)
  2. (4x^2 + 2x + 5x^2 + 6x + 40x + 8xy)

Solution

  1. (y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2)

Look at the variables and exponents. The expression contains (y^3, x^2, x), and constants. The terms (y^3) and (4y^3) are like terms because they both have (y^3). The terms (7x^2) and (5x^2) are like terms because they both have (x^2). The terms (14) and (23) are like terms because they are both constants. The term (9x) does not have any like terms in this list since no other terms have the variable (x) raised to the power of (1).

  1. (4x^2 + 2x + 5x^2 + 6x + 40x + 8xy)

Look at the variables and exponents. The expression contains the terms (4x^2, 2x, 5x^2, 6x, 40x), and (8xy) The terms (4x^2) and (5x^2) are like terms because they both have (x^2). The terms (2x, 6x), and (40x) are like terms because they all have (x). The term (8xy) has no like terms in the given expression because no other terms contain the two variables (xy).

exercise (PageIndex{15})

Identify the like terms in the list or the expression: (9, 2x^3, y^2, 8x^3, 15, 9y, 11y^2)

Answer

(9, 15); (2x^3) and (8x^3), (y^2), and (11y^2)

exercise (PageIndex{16})

Identify the like terms in the list or the expression: (4x^3 + 8x^2 + 19 + 3x^2 + 24 + 6x^3)

Answer

(4x^3) and (6x^3); (8x^2) and (3x^2); (19) and (24)

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think (3x + 6x) would simplify to? If you thought (9x), you would be right!

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what (x) is. If you have (3) of something and add (6) more of the same thing, the result is (9) of them. For example, (3) oranges plus (6) oranges is (9) oranges. We will discuss the mathematical properties behind this later.

The expression (3x + 6x) has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

Now it is easier to see the like terms to be combined.

HOW TO: COMBINE LIKE TERMS

Step 1. Identify like terms.

Step 2. Rearrange the expression so like terms are together.

Step 3. Add the coefficients of the like terms.

Example (PageIndex{9}): simplify

Simplify the expression: (3x + 7 + 4x + 5).

Solution

(3x + 7 + 4x + 5)
Identify the like terms( extcolor{red}{3x} + extcolor{blue}{7} + extcolor{red}{4x} + extcolor{blue}{5})
Rearrange the expression, so the like terms are together.( extcolor{red}{3x} + extcolor{red}{4x} + extcolor{blue}{7} + extcolor{blue}{5})
Add the coefficients of the like terms.( extcolor{red}{7x} + extcolor{blue}{12})
The original expression is simplified to...(7x + 12)

exercise (PageIndex{17})

Simplify: (7x + 9 + 9x + 8)

Answer

(16x+17)

exercise (PageIndex{18})

Simplify: (5y + 2 + 8y + 4y + 5)

Answer

(17y+7)

Example (PageIndex{10}): simplify

Simplify the expression: (7x^2 + 8x + x^2 + 4x).

Solution

(7x^{2} + 8x + x^{2} + 4x)
Identify the like terms.( extcolor{red}{7x^{2}} + extcolor{blue}{8x} + extcolor{red}{x^{2}} + extcolor{blue}{4x})
Rearrange the expression so like terms are together.( extcolor{red}{7x^{2}} + extcolor{red}{x^{2}} + extcolor{blue}{8x} + extcolor{blue}{4x})
Add the coefficients of the like terms.( extcolor{red}{8x^{2}} + extcolor{blue}{12x})

These are not like terms and cannot be combined. So (8x^2 + 12x) is in simplest form.

exercise (PageIndex{19})

Simplify: (3x^2 + 9x + x^2 + 5x)

Answer

(4x^2+14x)

exercise (PageIndex{20})

Simplify: (11y^2 + 8y + y^2 + 7y)

Answer

(12y^2+15y)


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10.3: Evaluate, Simplify, and Translate Expressions (Part 1)

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 +4x 2

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 +3x 2

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 -9x 2

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 -3x 2

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 -8x 2

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 +2x 2

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 +9x 2

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 +2x 2

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 -6x 2

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 +4x 2

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 +9x 2

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 +2x 2

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 -2x 2

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 -5x 2

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 +3x 2

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 -7x 2

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 -6x 2

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 +3x 2

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 -5x 2

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 +5x 2

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 -4x 2

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


Evaluating expressions practice

Evaluating expressions with multiple variables (practice), Practice evaluating expressions in two variables by plugging in values for the variables. Learn 4000+ math skills online. Get personalized guidance. Win fun awards!

Evaluating expressions with one variable (practice), Practice plugging in a value for the variable in an equation. These are introductory problems, so the expressions aren't too complicated. Practice evaluating expressions in two variables by plugging in values for the variables. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Evaluate expressions (Pre-Algebra, Introducing Algebra) – Mathplanet, Practice evaluating basic algebraic expressions. This interactive math game will help students gain a greater understanding of basic algebra. In this online math Evaluating expressions with one variable (practice) | Khan Academy. Practice plugging in a value for the variable in an equation. These are introductory problems, so the expressions aren't too complicated. Practice plugging in a value for the variable in an equation.


Algebraic Expressions Part 3

1. Translate "nine less than the total of a number and forty" into an algebraic expression, and simplify.

2. The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.

3. One hundred gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount g poured into the larger container.

3. I am not too sure about the wording here.

Subhotosh Khan

Super Moderator

1. Translate "nine less than the total of a number and forty" into an algebraic expression, and simplify.

2. The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.

3. One hundred gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount g poured into the larger container.

3. I am not too sure about the wording here.

3. One hundred gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount g poured into the larger container.


9 Answers 9

Macros are simply textual substitution, so in your example writing TIMER_100_MS in a program is a fancy way of writing 32768 / 10 .

Therefore, the question is when the compiler would evaluate 32768 / 10 , which is a constant integral expression. I don't think the standard requires any particular behavior here (since run-time and compile-time evaluation is indistinguishable in effect), but any halfway decent compiler will evaluate it at compile time.

Most answers in here focused on the effect of the macro substitution. But i think he wanted to know whether

is evaluated at compile time. First of all, that is an arithmetic constant expression, and in addition a integral constant expression (because it has only got literals of integer type). The implementation is free to calculate it at runtime, but it must also be able to calculate it at compile time, because

  1. it must give a diagnostic message if a constant expression is not representable in the type that its expression has
  2. such expressions are allowed in contexts that require the value at translation time, for example if used as the size of an array dimension.

If the compiler can principally calculate the result already at compile time, it should use that value, and not recalculate it at runtime i think. But maybe there is some reason still to do that. I wouldn't know.

Edit: I'm sorry i've answered the question as if it were about C++. Noticed today you asked about C. Overflowing in an expression is deemed as undefined behavior in C, regardless of whether it happens in a constant expression or not. The second point is also true in C, of course.

Edit: As a comment notes, if the macro is substituted into an expression like 3 * TIMER_100_MS , then this would evaluate (3 * 32768) / 10 . Therefore, the simple and direct answer is "No, it would not occur at runtime every time, because the division may not occur at all because of precedence and associativity rules". My answer above assumes that the macro is always substituted such that the division actually happens.


Floating-point arithmetic error

Floating-point error mitigation, Since most floating-point calculations have rounding error anyway, does it matter if the basic arithmetic operations introduce a little bit more rounding error than Error-analysis tells us how to design floating-point arithmetic, like IEEE Standard 754, moderately tolerant of well-meaning ignorance among programmers". [23] The special values such as infinity and NaN ensure that the floating-point arithmetic is algebraically completed, such that every floating-point operation produces a well-defined result and will not—by default—throw a machine interrupt or trap.

What Every Computer Scientist Should Know About Floating-Point , Explanations about propagation of errors in floating-point math. even simple calculations on them can contain pitfalls that increase the error in the result Scientist Should Know About Floating-Point Arithmetic gives a detailed introduction, Or If the result of an arithmetic operation gives a number smaller than .1000 E-99then it is called an underflow condition. Similarly, any result greater than .9999 E 99leads to an overflow condition. Floating Point Arithmetic. Floating point arithmetic is not associative.

Floating-point arithmetic, But you can “amplify” the representation error by repeatedly adding the Since many floating-point numbers are merely approximations of the exact That sort of thing is called Interval arithmetic and at least for me it was part Floating-point error mitigation is the minimization of errors caused by the fact that real numbers cannot, in general, be accurately represented in a fixed space. By definition, floating-point error cannot be eliminated, and, at best, can only be managed. H. M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the re


Expression calculator

Algebra Calculator, Simplifies expressions step-by-step and shows the work! To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. Free simplify calculator - simplify algebraic expressions step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy.

Simplify Calculator, Free equations calculator - solve linear, quadratic, polynomial, radical, exponential and logarithmic equations with all the steps. Type in any equation to get the Algebraic Expressions Calculator An online algebra calculator simplifies expression for the input you given in the input box. If you feel difficulty in solving some tough algebraic expression, this page will help you to solve the equation in a second.

Equation Calculator, Hint: Use the Equation Calculator for equations (containing = signs). Quick-Start Guide. When you enter an expression into the calculator, the calculator will When you enter an expression into the calculator, the calculator will simplify the expression by expanding multiplication and combining like terms. Use the following rules to enter expressions into the calculator.


Simplifying Rational Expressions

These lessons help Algebra and Grade 9 students learn how to simplify rational expressions.

Simplifying Rational Expressions

Students learn that when simplifying a rational expression, the first step is to factor both the numerator and denominator, and the next step is to cancel out the factors that match up,

Simplifying Rational Expressions Part 1

This lesson shows how to simplify rational expressions.

Simplifying Rational Expressions Part 2

This lesson shows how to simplify rational expressions (part 2).

Simplifying Rational Expressions

A rational expression means that there are variables on the bottom or top of a fraction. Factor the top part of a rational expression to cross cancel a part of the variable

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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