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Algebraic Expressions

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses algebraic expressions. By the end of the module students should be able to recognize an algebraic expression, be able to distinguish between terms and factors, understand the meaning and function of coefficients and be able to perform numerical evaluation.

Section Overview

  • Algebraic Expressions
  • Terms and Factors
  • Coefficients
  • Numerical Evaluation

Algebraic Expressions

Numerical Expression

In arithmetic, a numerical expression results when numbers are connected by arithmetic operation signs (+, -, ⋅ , ÷). For example, 8+58+5 size 12{8+5} {}, 4−94−9 size 12{4 - 9} {}, 3⋅83⋅8 size 12{3 cdot 8} {}, and 9÷79÷7 size 12{9 div 7} {} are numerical expressions.

Algebraic Expression

In algebra, letters are used to represent numbers, and an algebraic expression results when an arithmetic operation sign associates a letter with a number or a letter with a letter. For example, x+8x+8 size 12{x+8} {}, 4−y4−y size 12{4 - y} {} , 3⋅x3⋅x size 12{3 cdot x} {} , x÷7x÷7 size 12{x div 7} {} , and x⋅yx⋅y size 12{x cdot y} {} are algebraic expressions.

Expressions

Numerical expressions and algebraic expressions are often referred to simply as expressions.

Terms and Factors

In algebra, it is extremely important to be able to distinguish between terms and factors.

Distinction Between Terms and Factors

Terms are parts of sums and are therefore connected by + signs.
Factors are parts of products and are therefore separated by ⋅ signs.

Note:

While making the distinction between sums and products, we must re­member that subtraction and division are functions of these operations.
  1. In some expressions it will appear that terms are separated by minus signs. We must keep in mind that subtraction is addition of the opposite, that is,
    x − y = x + ( − y ) x − y = x + ( − y ) size 12{x - y=x+ \( - y \) } {}
  2. In some expressions it will appear that factors are separated by division signs. We must keep in mind that
    x y = x 1 ⋅ 1 y = x ⋅ 1 y x y = x 1 ⋅ 1 y = x ⋅ 1 y size 12{ { {x} over {y} } = { {x} over {1} } cdot { {1} over {y} } =x cdot { {1} over {y} } } {}

Sample Set A

State the number of terms in each expression and name them.

Example 1

x+4x+4 size 12{x+4} {}. In this expression, x and 4 are connected by a "+" sign. Therefore, they are terms. This expression consists of two terms.

Example 2

y−8y−8 size 12{y - 8} {}. The expression y−8y−8 size 12{y - 8} {} can be expressed as y+(−8)y+(−8) size 12{y+ \( - 8 \) } {}. We can now see that this expres­sion consists of the two terms yy size 12{y} {} and −8−8 size 12{ - 8} {}.

Rather than rewriting the expression when a subtraction occurs, we can identify terms more quickly by associating the + or - sign with the individual quantity.

Example 3

a+7−b−ma+7−b−m size 12{a+7 - b - m} {}. Associating the sign with the individual quantities, we see that this expression consists of the four terms aa size 12{a} {}, 7, −b−b size 12{ - b} {}, −m−m size 12{ - m} {}.

Example 4

5 m−8 n5 m−8 n size 12{5m - 8n} {}. This expression consists of the two terms, 5 m5 m size 12{5m} {} and −8 n−8 n size 12{ - 8n} {}. Notice that the term 5 m5 m size 12{5m} {} is composed of the two factors 5 and mm size 12{m} {}. The term −8 n−8 n size 12{ - 8n} {} is composed of the two factors −8−8 size 12{ - 8} {} and nn size 12{n} {}.

Example 5

3 x3 x size 12{3x} {}. This expression consists of one term. Notice that 3 x3 x size 12{3x} {} can be expressed as 3 x+03 x+0 size 12{3x+0} {} or 3 x⋅13 x⋅1 size 12{3x cdot 1} {} (indicating the connecting signs of arithmetic). Note that no operation sign is necessary for multiplication.

Practice Set A

Specify the terms in each expression.

Exercise 1

x+7x+7 size 12{x+7} {}

Solution

x x , 7

Exercise 2

3 m−6 n3 m−6 n size 12{3m - 6n} {}

Solution

3 m−6 n3 m−6 n size 12{3m - 6n} {}

Exercise 3

5 y5 y size 12{5y} {}

Solution

5 y5 y size 12{5y} {}

Exercise 4

a+2 b−ca+2 b−c size 12{a+2b - c} {}

Solution

a , 2 b ,−ca , 2 b ,−c size 12{"a, 2b, "-c} {}

Exercise 5

−3 x−5−3 x−5 size 12{ - 3x - 5} {}

Solution

−3 x,−5−3 x,−5 size 12{-3x,-5} {}

Coefficients

We know that multiplication is a description of repeated addition. For example,
5⋅75⋅7 size 12{5 cdot 7} {} describes 7+7+7+7+77+7+7+7+7 size 12{7+7+7+7+7} {}

Suppose some quantity is represented by the letter xx size 12{x} {}. The multiplication 5x5x size 12{5x} {} de­scribes x+x+x+x+xx+x+x+x+x size 12{x+x+x+x+x} {}. It is now easy to see that 5 x5 x size 12{5x} {} specifies 5 of the quantities represented by xx size 12{x} {}. In the expression 5 x5 x size 12{5x} {}, 5 is called the numerical coefficient, or more simply, the coefficient of xx size 12{x} {}.

Coefficient

The coefficient of a quantity records how many of that quantity there are.

Since constants alone do not record the number of some quantity, they are not usually considered as numerical coefficients. For example, in the expression 7 x+2 y−8 z+127 x+2 y−8 z+12 size 12{7x+2y - 8z+"12"} {}, the coefficient of

7 x7 x size 12{7x} {} is 7. (There are 7 x's.)
2 y2 y size 12{2y} {} is 2. (There are 2 y 's.)
−8 z−8 z size 12{ - 8z} {} is −8−8 size 12{ - 8} {}. (There are −8−8 size 12{ - 8} {}z 's.)

The constant 12 is not considered a numerical coefficient.

1x=x1x=x

When the numerical coefficient of a variable is 1, we write only the variable and not the coefficient. For example, we write xx size 12{x} {} rather than 1 x1 x size 12{1x} {}. It is clear just by looking at xx size 12{x} {} that there is only one.

Numerical Evaluation

We know that a variable represents an unknown quantity. Therefore, any expres­sion that contains a variable represents an unknown quantity. For example, if the value of xx size 12{x} {} is unknown, then the value of 3 x+53 x+5 size 12{3x+5} {} is unknown. The value of 3 x+53 x+5 size 12{3x+5} {} depends on the value of xx size 12{x} {}.

Numerical Evaluation

Numerical evaluation is the process of determining the numerical value of an algebraic expression by replacing the variables in the expression with specified numbers.

Sample Set B

Find the value of each expression.

Example 6

2 x+7 y2 x+7 y size 12{2x+7y} {}, if x = - 4 x = - 4 and y=2y=2 size 12{y=2} {}

Replace x with –4 and y with 2.

2x+7y = 2(- 4)+7(2) = - 8+14 = 6 2x+7y = 2(- 4)+7(2) = - 8+14 = 6

Thus, when x = – 4 x = – 4 and y = 2 y = 2, 2 x + 7 y= 62 x + 7 y= 6 size 12{2x+7y=6} {}.

Example 7

5a b + 8b 12 5a b + 8b 12 , if a=6a=6 and bb = - 3- 3.

Replace a with 6 and b with –3.

5a b + 8b 12 = 5(6) - 3 + 8(- 3) 12 = 30 - 3 + - 24 12 = - 10+(- 2) = - 12 5a b + 8b 12 = 5(6) - 3 + 8(- 3) 12 = 30 - 3 + - 24 12 = - 10+(- 2) = - 12

Thus, when a = 6 and b = –3, 5ab+8b12=-125ab+8b12=-12 size 12{ { {5a} over {b} } + { {8b} over {"12"} } "=-""12"} {}.

Example 8

62 a−15b62 a−15b size 12{6 left (2a-"15"b right )} {}, if a=-5a=-5 size 12{a"=-"5} {} and b=-1b=-1 size 12{b"=-"1} {}

Replace a a with –5 and b b with –1.

6(2a-15b) = 6(2(- 5)-15(- 1)) = 6(- 10+15) = 6(5) = 30 6(2a-15b) = 6(2(- 5)-15(- 1)) = 6(- 10+15) = 6(5) = 30

Thus, when a = –5 a = –5 and b= –1 b= –1 , 62 a−15b=3062 a−15b=30 size 12{6 left (2a-"15"b right )="30"} {}.

Example 9

3 x2−2 x+13 x2−2 x+1 size 12{3x rSup { size 8{2} } -2x+1} {}, if x=4x=4 size 12{x=4} {}

Replace x with 4.

3x2-2x+1 = 3(4)2-2(4)+1 = 3⋅16-2(4)+1 = 48-8+1 = 41 3x2-2x+1 = 3(4)2-2(4)+1 = 3⋅16-2(4)+1 = 48-8+1 = 41

Thus, when x = 4 x = 4 , 3 x2−2 x+1=41 3 x2−2 x+1=41 size 12{3x rSup { size 8{2} } -2x+1="41"} {}.

Example 10

−x2−4−x2−4 size 12{-x rSup { size 8{2} } -4} {}, if x=3x=3 size 12{x=3} {}

Replace x with 3.

-x2-4 = - 32-4 Be careful to square only the 3. The exponent 2 is connected  only   to 3, not -3 = - 9-4 = - 13 -x2-4 = - 32-4 Be careful to square only the 3. The exponent 2 is connected  only   to 3, not -3 = - 9-4 = - 13

Example 11

−x2−4−x2−4 size 12{ left (-x right ) rSup { size 8{2} } -4} {}, if x=3x=3 size 12{x=3} {}.

Replace x with 3.

(-x)2-4 = (- 3)2-4 The exponent is connected to -3, not 3 as in problem 5 above. = 9-4 = - 5 (-x)2-4 = (- 3)2-4 The exponent is connected to -3, not 3 as in problem 5 above. = 9-4 = - 5

The exponent is connected to –3, not 3 as in the problem above.

Practice Set B

Find the value of each expression.

Exercise 6

9 m−2 n,9 m−2 n, size 12{9m-2n,} {} if m=-2m=-2 size 12{m"=-"2} {} and n=5n=5 size 12{n=5} {}

Solution

-28

Exercise 7

−3 x − 5 y + 2 z −3 x − 5 y + 2 z size 12{-3x-5y+2z} {}, if x=-4x=-4 size 12{x"=-"4} {}, y=3y=3 size 12{y=3} {}, z=0z=0 size 12{z=0} {}

Solution

-3

Exercise 8

10a3b+4b210a3b+4b2 size 12{ { {"10"a} over {3b} } + { {4b} over {2} } } {}, if a=-6a=-6 size 12{a"=-"6} {}, and b=2b=2 size 12{b=2} {}

Solution

-6

Exercise 9

83 m−5 n83 m−5 n size 12{8 left (3m-5n right )} {}, if m=-4m=-4 size 12{m"=-"4} {} and n=-5n=-5 size 12{n"=-"5} {}

Solution

104

Exercise 10

3−40−24 a−3 b3−40−24 a−3 b size 12{3 left [-"40"-2 left (4a-3b right ) right ]} {}, if a=-6a=-6 size 12{a"=-"6} {} and b=0b=0 size 12{b=0} {}

Solution

24

Exercise 11

5 y2+6 y−115 y2+6 y−11 size 12{5y rSup { size 8{2} } +6y-"11"} {}, if y=-1y=-1 size 12{y"=-"1} {}

Solution

-12

Exercise 12

−x2+2 x+7−x2+2 x+7 size 12{-x rSup { size 8{2} } +2x+7} {}, if x=4x=4 size 12{x=4} {}

Solution

-1

Exercise 13

−x2+2 x+7−x2+2 x+7 size 12{ left (-x right ) rSup { size 8{2} } +2x+7} {}, if x=4x=4 size 12{x=4} {}

Solution

31

Exercises

Exercise 14

In an algebraic expression, terms are separated by
signs and factors are separated by
signs.

Solution

Addition; multiplication

For the following 8 problems, specify each term.

Exercise 15

3 m+7 n3 m+7 n size 12{3m+7n} {}

Exercise 16

5 x+18y5 x+18y size 12{5x+"18"y} {}

Solution

5 x, 18 y5 x, 18 y size 12{5x, 18y} {}

Exercise 17

4 a−6 b+c4 a−6 b+c size 12{4a-6b+c} {}

Exercise 18

8 s+2 r−7 t8 s+2 r−7 t size 12{8s+2r-7t} {}

Solution

8 s, 2 r, −7 t8 s, 2 r, −7 t size 12{8s,2r,-7t} {}

Exercise 19

m−3 n−4 a+7 bm−3 n−4 a+7 b size 12{m-3n-4a+7b} {}

Exercise 20

7 a−2 b−3 c−4 d7 a−2 b−3 c−4 d size 12{7a-2b-3c-4d} {}

Solution

7 a,−2 b, −3 c, −4 d7 a,−2 b, −3 c, −4 d size 12{7a, -2b, -3c, -4d} {}

Exercise 21

−6 a−5 b−6 a−5 b size 12{-6a-5b} {}

Exercise 22

−x−y−x−y size 12{-x-y} {}

Solution

−x,−y−x,−y size 12{-x,-y} {}

Exercise 23

What is the function of a numerical coefficient?

Exercise 24

Write 1 m 1 m in a simpler way.

Solution

m m

Exercise 25

Write 1s in a simpler way.

Exercise 26

In the expression 5a, how many a’s are indicated?

Solution

5

Exercise 27

In the expression –7c, how many c’s are indicated?

Find the value of each expression.

Exercise 28

2 m−6 n2 m−6 n size 12{2m-6n} {}, if m=-3m=-3 size 12{m"=-"3} {} and n=4n=4 size 12{n=4} {}

Solution

-30

Exercise 29

5 a+6 b5 a+6 b size 12{5a+6b} {}, if a=-6a=-6 size 12{a"=-"6} {} and b=5b=5 size 12{b=5} {}

Exercise 30

2 x − 3 y+ 4 z2 x − 3 y+ 4 z size 12{2x-3y+4z} {}, if x=1x=1 size 12{x=1} {}, y=-1y=-1 size 12{y"=-"1} {}, and z=-2z=-2 size 12{z"=-"2} {}

Solution

-3

Exercise 31

9 a+6 b−8 x+4 y9 a+6 b−8 x+4 y size 12{9a+6b-8x+4y} {}, if a=-2a=-2 size 12{a"=-"2} {}, b=-1b=-1 size 12{b"=-"1} {}, x=-2x=-2 size 12{x"=-"2} {}, and y=0y=0 size 12{y=0} {}

Exercise 32

8 x 3 y +18y 2 x , 8 x 3 y +18y 2 x , size 12{ { {8x} over {3y} } + { {"18"y} over {2x} } ,} {} if x=9x=9 size 12{x=9} {} and y=-2y=-2 size 12{y"=-"2} {}

Solution

-14

Exercise 33

−3 m 2 n −−6 nm,−3 m 2 n −−6 nm, size 12{ { {-3m} over {2n} } - { {-6n} over {m} } ,} {} if m=-6m=-6 size 12{m"=-"6} {} and n=3n=3 size 12{n=3} {}

Exercise 34

43 r+2 s43 r+2 s size 12{4 left (3r+2s right )} {}, if r=4r=4 size 12{r=4} {} and s=1s=1 size 12{s=1} {}

Solution

56

Exercise 35

39 a−6 b39 a−6 b size 12{3 left (9a-6b right )} {}, if a=-1a=-1 size 12{a"=-"1} {} and b=-2b=-2 size 12{b"=-"2} {}

Exercise 36

−85 m+8 n−85 m+8 n size 12{-8 left (5m+8n right )} {}, if m=0m=0 size 12{m=0} {} and n=-1n=-1 size 12{n"=-"1} {}

Solution

64

Exercise 37

−2−6 x+y−2 z−2−6 x+y−2 z size 12{-2 left (-6x+y-2z right )} {}, if x=1x=1 size 12{x=1} {}, y=1y=1 size 12{y=1} {}, and z=2z=2 size 12{z=2} {}

Exercise 38

−10x−2 y+5 z−10x−2 y+5 z size 12{- left ("10"x-2y+5z right )} {} if x=2x=2 size 12{x=2} {}, y=8y=8 size 12{y=8} {}, and z=-1z=-1 size 12{z"=-"1} {}

Solution

1

Exercise 39

−a−3 b+2 c−d−a−3 b+2 c−d size 12{- left (a-3b+2c-d right )} {}, if a=-5a=-5 size 12{a"=-"5} {}, b=2b=2 size 12{b=2} {}, c=0c=0 size 12{c=0} {}, and d=-1d=-1 size 12{d"=-"1} {}

Exercise 40

316−3a+3 b316−3a+3 b size 12{3 left ["16"-3 left (a+3b right ) right ]} {}, if a=3a=3 size 12{a=3} {} and b=-2b=-2 size 12{b"=-"2} {}

Solution

75

Exercise 41

−25 a+2 bb−6−25 a+2 bb−6 size 12{-2 left [5a+2b left (b-6 right ) right ]} {}, if a=-2a=-2 size 12{a"=-"2} {} and b=3b=3 size 12{b=3} {}

Exercise 42

−6 x+3 y−2x+4 y−6 x+3 y−2x+4 y size 12{- left lbrace 6x+3y left [-2 left (x+4y right ) right ] right rbrace } {}, if x=0x=0 size 12{x=0} {} and y=1y=1 size 12{y=1} {}

Solution

24

Exercise 43

−219−64−2a−b−7−219−64−2a−b−7 size 12{-2 left lbrace "19"-6 left [4-2 left (a-b-7 right ) right ] right rbrace } {}, if a=10a=10 size 12{a="10"} {} and b=3b=3 size 12{b=3} {}

Exercise 44

x2+3 x−1x2+3 x−1 size 12{x rSup { size 8{2} } +3x-1} {}, if x=5x=5 size 12{x=5} {}

Solution

39

Exercise 45

m2−2 m+6m2−2 m+6 size 12{m rSup { size 8{2} } -2m+6} {}, if m=3m=3 size 12{m=3} {}

Exercise 46

6 a2+2 a−156 a2+2 a−15 size 12{6a rSup { size 8{2} } +2a-"15"} {}, if a=-2a=-2 size 12{a"=-"2} {}

Solution

5

Exercise 47

5 s2+6 s+10,5 s2+6 s+10, size 12{5s rSup { size 8{2} } +6s+"10",} {} if x=-1x=-1 size 12{x"=-"1} {}

Exercise 48

16x2+8 x−716x2+8 x−7 size 12{"16"x rSup { size 8{2} } +8x-7} {}, if x=0x=0 size 12{x=0} {}

Solution

-7

Exercise 49

−8 y2+6 y+11,−8 y2+6 y+11, size 12{-8y rSup { size 8{2} } +6y+"11",} {} if y=0y=0 size 12{y=0} {}

Exercise 50

y−62+3y−5+4y−62+3y−5+4 size 12{ left (y-6 right ) rSup { size 8{2} } +3 left (y-5 right )+4} {}, if y=5y=5 size 12{y=5} {}

Solution

5

Exercise 51

x+82+4x+9+1,x+82+4x+9+1, size 12{ left (x+8 right ) rSup { size 8{2} } +4 left (x+9 right )+1,} {} if x=-6x=-6 size 12{x"=-"6} {}

Exercises for Review

Exercise 52

((Reference)) Perform the addition: 538+216538+216 size 12{5 { {3} over {8} } +2 { {1} over {6} } } {}.

Solution

18124=7132418124=71324 size 12{ { {"181"} over {"24"} } =7 { {"13"} over {"24"} } } {}

Exercise 53

((Reference)) Arrange the numbers in order from smallest to largest: 1132,1548, and 7161132,1548, and 716 size 12{ { {"11"} over {"32"} } , { {"15"} over {"48"} } ", and " { {7} over {"16"} } } {}

Exercise 54

((Reference)) Find the value of 232+827232+827 size 12{ left ( { {2} over {3} } right ) rSup { size 8{2} } + { {8} over {"27"} } } {}

Solution

20272027 size 12{ { {"20"} over {"27"} } } {}

Exercise 55

((Reference)) Write the proportion in fractional form: “9 is to 8 as x is to 7.”

Exercise 56

((Reference)) Find the value of −32−6−12−32−6−12 size 12{-3 left (2-6 right )-"12"} {}

Solution

0

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