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Algebraic Expressions and Equations: Combining Polynomials Using Addition and Subtraction

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form. Objectives of this module: understand the concept of like terms, be able to combine like terms, be able to simplify expressions containing parentheses.

Overview

• Like Terms
• Combining Like Terms
• Simplifying Expressions Containing Parentheses

Like Terms

Like Terms

Terms whose variable parts, including the exponents, are identical are called like terms. Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. As long as we are dealing with quantities of the same type we can combine them using addition and subtraction.

Simplifying an Algebraic Expression

An algebraic expression can be simplified by combining like terms.

Sample Set A

Combine the like terms.

Example 1

6houses+4houses=10houses 6houses+4houses=10houses . 6 and 4 of the same type give 10 of that type.

Example 2

6houses+4houses+2motels=10houses+2motels 6houses+4houses+2motels=10houses+2motels . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type.

Example 3

Suppose we let the letter x x represent "house." Then, 6x+4x=10x 6x+4x=10x . 6 and 4 of the same type give 10 of that type.

Example 4

Suppose we let x x represent "house" and y y represent "motel."

6x+4x+2y=10x+2y 6x+4x+2y=10x+2y
(1)

Practice Set A

Like terms with the same numerical coefficient represent equal amounts of the same quantity.

Exercise 1

Like terms with different numerical coefficients represent


.

Solution

different amounts of the same quantity

Combining Like Terms

Since like terms represent amounts of the same quantity, they may be combined, that is, like terms may be added together.

Sample Set B

Simplify each of the following polynomials by combining like terms.

Example 5

2x+5x+3x 2x+5x+3x .
There are 2x's 2x's , then 5 more, then 3 more. This makes a total of 10x's 10x's .

2x+5x+3x=10x 2x+5x+3x=10x

Example 6

7x+8y3x 7x+8y3x .
From 7x's 7x's , we lose 3x's 3x's . This makes 4x's 4x's . The 8y's 8y's represent a quantity different from the x's x's and therefore will not combine with them.

7x+8y3x=4x+8y 7x+8y3x=4x+8y

Example 7

4 a 3 2 a 2 +8 a 3 + a 2 2 a 3 4 a 3 2 a 2 +8 a 3 + a 2 2 a 3 .
4 a 3 ,8 a 3 , 4 a 3 ,8 a 3 , and 2 a 3 2 a 3 represent quantities of the same type.

4 a 3 +8 a 3 2 a 3 =10 a 3 4 a 3 +8 a 3 2 a 3 =10 a 3

2 a 2 2 a 2 and a 2 a 2 represent quantities of the same type.

2 a 2 + a 2 = a 2 2 a 2 + a 2 = a 2

Thus,

4 a 3 2 a 2 +8 a 3 + a 2 2 a 3 =10 a 3 a 2 4 a 3 2 a 2 +8 a 3 + a 2 2 a 3 =10 a 3 a 2

Practice Set B

Simplify each of the following expressions.

4y+7y 4y+7y

11y 11y

Exercise 3

3x+6x+11x 3x+6x+11x

20x 20x

Exercise 4

5a+2b+4ab7b 5a+2b+4ab7b

9a6b 9a6b

Exercise 5

10 x 3 4 x 3 +3 x 2 12 x 3 +5 x 2 +2x+ x 3 +8x 10 x 3 4 x 3 +3 x 2 12 x 3 +5 x 2 +2x+ x 3 +8x

Solution

5 x 3 +8 x 2 +10x 5 x 3 +8 x 2 +10x

Exercise 6

2 a 5 a 5 +14ab9+9ab23 a 5 2 a 5 a 5 +14ab9+9ab23 a 5

5ab13 5ab13

Simplifying Expressions Containing Parentheses

Simplifying Expressions Containing Parentheses

When parentheses occur in expressions, they must be removed before the expression can be simplified. Parentheses can be removed using the distributive property.

Sample Set C

Simplify each of the following expressions by using the distributive property and combining like terms.

Example 9

4x+9( x 2 6x2)+5 Removeparentheses. 4x+9 x 2 54x18+5 Combineliketerms. 50x+9 x 2 13 4x+9( x 2 6x2)+5 Removeparentheses. 4x+9 x 2 54x18+5 Combineliketerms. 50x+9 x 2 13

By convention, the terms in an expression are placed in descending order with the highest degree term appearing first. Numerical terms are placed at the right end of the expression. The commutative property of addition allows us to change the order of the terms.

9 x 2 50x13 9 x 2 50x13

Example 10

2+2[5+4(1+a)] 2+2[5+4(1+a)]
Eliminate the innermost set of parentheses first.

2+2[5+4+4a] 2+2[5+4+4a]

By the order of operations, simplify inside the parentheses before multiplying (by the 2).

2+2[9+4a] Removethissetofparentheses. 2+18+8a Combineliketerms. 20+8a Writeindescendingorder. 8a+20 2+2[9+4a] Removethissetofparentheses. 2+18+8a Combineliketerms. 20+8a Writeindescendingorder. 8a+20

Example 11

x(x3)+6x(2x+3) x(x3)+6x(2x+3)
Use the rule for multiplying powers with the same base.

x 2 3x+12 x 2 +18x Combineliketerms. 13 x 2 +15x x 2 3x+12 x 2 +18x Combineliketerms. 13 x 2 +15x

Practice Set C

Simplify each of the following expressions by using the distributive property and combining like terms.

Exercise 7

4(x+6)+3(2+x+3 x 2 )2 x 2 4(x+6)+3(2+x+3 x 2 )2 x 2

Solution

7 x 2 +7x+30 7 x 2 +7x+30

Exercise 8

7(x+ x 3 )4 x 3 x+1+4( x 2 2 x 3 +7) 7(x+ x 3 )4 x 3 x+1+4( x 2 2 x 3 +7)

Solution

5 x 3 +4 x 2 +6x+29 5 x 3 +4 x 2 +6x+29

Exercise 9

5(a+2)+6a7+(8+4)(a+3a+2) 5(a+2)+6a7+(8+4)(a+3a+2)

59a+27 59a+27

Exercise 10

x(x+3)+4 x 2 +2x x(x+3)+4 x 2 +2x

Solution

5 x 2 +5x 5 x 2 +5x

Exercise 11

a 3 ( a 2 +a+5)+a( a 4 +3 a 2 +4)+1 a 3 ( a 2 +a+5)+a( a 4 +3 a 2 +4)+1

Solution

2 a 5 + a 4 +8 a 3 +4a+1 2 a 5 + a 4 +8 a 3 +4a+1

Exercise 12

2[83(x3)] 2[83(x3)]

6x+34 6x+34

Exercise 13

x 2 +3x+7[x+4 x 2 +3(x+ x 2 )] x 2 +3x+7[x+4 x 2 +3(x+ x 2 )]

Solution

50 x 2 +31x 50 x 2 +31x

Exercises

For the following problems, simplify each of the algebraic expressions.

x+3x x+3x

4x 4x

4x+7x 4x+7x

9a+12a 9a+12a

21a 21a

5m3m 5m3m

10x7x 10x7x

3x 3x

7y9y 7y9y

6k11k 6k11k

5k 5k

Exercise 21

3a+5a+2a 3a+5a+2a

Exercise 22

9y+10y+2y 9y+10y+2y

21y 21y

5m7m2m 5m7m2m

h3h5h h3h5h

7h 7h

a+8a+3a a+8a+3a

7ab+4ab 7ab+4ab

11ab 11ab

Exercise 27

8ax+2ax+6ax 8ax+2ax+6ax

Exercise 28

3 a 2 +6 a 2 +2 a 2 3 a 2 +6 a 2 +2 a 2

11 a 2 11 a 2

Exercise 29

14 a 2 b+4 a 2 b+19 a 2 b 14 a 2 b+4 a 2 b+19 a 2 b

10y15y 10y15y

5y 5y

Exercise 31

7ab9ab+4ab 7ab9ab+4ab

Exercise 32

210a b 4 +412a b 4 +100 a 4 b (Lookcloselyattheexponents.) 210a b 4 +412a b 4 +100 a 4 b (Lookcloselyattheexponents.)

Solution

622a b 4 +100 a 4 b 622a b 4 +100 a 4 b

Exercise 33

5 x 2 y 0 +3 x 2 y+2 x 2 y+1, y0 (Lookcloselyattheexponents.) 5 x 2 y 0 +3 x 2 y+2 x 2 y+1, y0 (Lookcloselyattheexponents.)

Exercise 34

8 w 2 12 w 2 3 w 2 8 w 2 12 w 2 3 w 2

7 w 2 7 w 2

Exercise 35

6xy3xy+7xy18xy 6xy3xy+7xy18xy

Exercise 36

7 x 3 2 x 2 10x+15 x 2 3 x 3 12+x 7 x 3 2 x 2 10x+15 x 2 3 x 3 12+x

Solution

4 x 3 7 x 2 9x11 4 x 3 7 x 2 9x11

Exercise 37

21y15x+40xy611y+712xxy 21y15x+40xy611y+712xxy

Exercise 38

1x+1y1x1y+xy 1x+1y1x1y+xy

xy xy

Exercise 39

5 x 2 3x7+2 x 2 x 5 x 2 3x7+2 x 2 x

Exercise 40

2 z 3 +15z+4 z 3 + z 2 6 z 2 +z 2 z 3 +15z+4 z 3 + z 2 6 z 2 +z

Solution

2 z 3 5 z 2 +16z 2 z 3 5 z 2 +16z

Exercise 41

18 x 2 y14 x 2 y20 x 2 y 18 x 2 y14 x 2 y20 x 2 y

Exercise 42

9 w 5 9 w 4 9 w 5 +10 w 4 9 w 5 9 w 4 9 w 5 +10 w 4

Solution

18 w 5 + w 4 18 w 5 + w 4

Exercise 43

2 x 4 +4 x 3 8 x 2 +12x17 x 3 1 x 4 6x+2 2 x 4 +4 x 3 8 x 2 +12x17 x 3 1 x 4 6x+2

Exercise 44

17 d 3 r+3 d 3 r5 d 3 r+6 d 2 r+ d 3 r30 d 2 r+37+2 17 d 3 r+3 d 3 r5 d 3 r+6 d 2 r+ d 3 r30 d 2 r+37+2

Solution

16 d 3 r24 d 2 r2 16 d 3 r24 d 2 r2

Exercise 45

a 0 +2 a 0 4 a 0 , a0 a 0 +2 a 0 4 a 0 , a0

Exercise 46

4 x 0 +3 x 0 5 x 0 +7 x 0 x 0 , x0 4 x 0 +3 x 0 5 x 0 +7 x 0 x 0 , x0

8

Exercise 47

2 a 3 b 2 c+3 a 2 b 2 c 0 +4 a 2 b 2 a 3 b 2 c, c0 2 a 3 b 2 c+3 a 2 b 2 c 0 +4 a 2 b 2 a 3 b 2 c, c0

3z6z+8z 3z6z+8z

5z 5z

Exercise 49

3 z 2 z+3 z 3 3 z 2 z+3 z 3

Exercise 50

6 x 3 +12x+5 6 x 3 +12x+5

Solution

6 x 3 +12x+5 6 x 3 +12x+5

Exercise 51

3(x+5)+2x 3(x+5)+2x

Exercise 52

7(a+2)+4 7(a+2)+4

7a+18 7a+18

Exercise 53

y+5(y+6) y+5(y+6)

Exercise 54

2b+6(35b) 2b+6(35b)

28b+18 28b+18

Exercise 55

5a7c+3(ac) 5a7c+3(ac)

Exercise 56

8x3x+4(2x+5)+3(6x4) 8x3x+4(2x+5)+3(6x4)

31x+8 31x+8

Exercise 57

2z+4ab+5zab+12(1abz) 2z+4ab+5zab+12(1abz)

Exercise 58

(a+5)4+6a20 (a+5)4+6a20

10a 10a

Exercise 59

(4a+5b2)3+3(4a+5b2) (4a+5b2)3+3(4a+5b2)

Exercise 60

(10x+3 y 2 )4+4(10x+3 y 2 ) (10x+3 y 2 )4+4(10x+3 y 2 )

Solution

80x+24 y 2 80x+24 y 2

2(x6)+5 2(x6)+5

Exercise 62

1(3x+15)+2x12 1(3x+15)+2x12

5x+3 5x+3

Exercise 63

1(2+9a+4 a 2 )+ a 2 11a 1(2+9a+4 a 2 )+ a 2 11a

Exercise 64

1(2x6b+6 a 2 b+8 b 2 )+1(5x+2b3 a 2 b) 1(2x6b+6 a 2 b+8 b 2 )+1(5x+2b3 a 2 b)

Solution

3 a 2 b+8 b 2 4b+7x 3 a 2 b+8 b 2 4b+7x

Exercise 65

After observing the following problems, can you make a conjecture about 1(a+b) 1(a+b) ?
1(a+b)= 1(a+b)=

Exercise 66

Using the result of problem 52, is it correct to write
(a+b)=a+b? (a+b)=a+b?

yes

Exercise 67

3(2a+2 a 2 )+8(3a+3 a 2 ) 3(2a+2 a 2 )+8(3a+3 a 2 )

Exercise 68

x(x+2)+2( x 2 +3x4) x(x+2)+2( x 2 +3x4)

Solution

3 x 2 +8x8 3 x 2 +8x8

Exercise 69

A(A+7)+4( A 2 +3a+1) A(A+7)+4( A 2 +3a+1)

Exercise 70

b(2 b 3 +5 b 2 +b+6)6 b 2 4b+2 b(2 b 3 +5 b 2 +b+6)6 b 2 4b+2

Solution

2 b 4 +5 b 3 5 b 2 +2b+2 2 b 4 +5 b 3 5 b 2 +2b+2

Exercise 71

4aa(a+5) 4aa(a+5)

Exercise 72

x3x( x 2 7x1) x3x( x 2 7x1)

Solution

3 x 3 +21 x 2 +4x 3 x 3 +21 x 2 +4x

Exercise 73

ab(a5)4 a 2 b+2ab2 ab(a5)4 a 2 b+2ab2

Exercise 74

xy(3xy+2x5y)2 x 2 y 2 5 x 2 y+4x y 2 xy(3xy+2x5y)2 x 2 y 2 5 x 2 y+4x y 2

Solution

x 2 y 2 3 x 2 yx y 2 x 2 y 2 3 x 2 yx y 2

Exercise 75

3h[2h+5(h+2)] 3h[2h+5(h+2)]

Exercise 76

2k[5k+3(1+7k)] 2k[5k+3(1+7k)]

Solution

52 k 2 +6k 52 k 2 +6k

Exercise 77

8a[2a4ab+9(a5ab)] 8a[2a4ab+9(a5ab)]

Exercise 78

6{m+5n[n+3(n1)]+2 n 2 }4 n 2 9m 6{m+5n[n+3(n1)]+2 n 2 }4 n 2 9m

Solution

128 n 2 90n3m 128 n 2 90n3m

Exercise 79

5[4(r2s)3r5s]+12s 5[4(r2s)3r5s]+12s

Exercise 80

8{9[b2a+6c(c+4)4 c 2 ]+4a+b}3b 8{9[b2a+6c(c+4)4 c 2 ]+4a+b}3b

Solution

144 c 2 112a+77b+1728c 144 c 2 112a+77b+1728c

Exercise 81

5[4(6x3)+x]2x25x+4 5[4(6x3)+x]2x25x+4

Exercise 82

3x y 2 (4xy+5y)+2x y 3 +6 x 2 y 3 +4 y 3 12x y 3 3x y 2 (4xy+5y)+2x y 3 +6 x 2 y 3 +4 y 3 12x y 3

Solution

18 x 2 y 3 +5x y 3 +4 y 3 18 x 2 y 3 +5x y 3 +4 y 3

Exercise 83

9 a 3 b 7 ( a 3 b 5 2 a 2 b 2 +6)2a( a 2 b 7 5 a 5 b 12 +3 a 4 b 9 ) a 3 b 7 9 a 3 b 7 ( a 3 b 5 2 a 2 b 2 +6)2a( a 2 b 7 5 a 5 b 12 +3 a 4 b 9 ) a 3 b 7

8(3a+2) 8(3a+2)

24a16 24a16

4(2x3y) 4(2x3y)

Exercise 86

4x y 2 [7xy6(5x y 2 )+3(xy+1)+1] 4x y 2 [7xy6(5x y 2 )+3(xy+1)+1]

Solution

24 x 2 y 4 16 x 2 y 3 +104x y 2 24 x 2 y 4 16 x 2 y 3 +104x y 2

Exercises for Review

Exercise 87

((Reference)) Simplify ( x 10 y 8 z 2 x 2 y 6 ) 3 ( x 10 y 8 z 2 x 2 y 6 ) 3 .

Exercise 88

((Reference)) Find the value of 3(49)6(3)1 2 3 3(49)6(3)1 2 3 .

4

Exercise 89

((Reference)) Write the expression 42 x 2 y 5 z 3 21 x 4 y 7 42 x 2 y 5 z 3 21 x 4 y 7 so that no denominator appears.

Exercise 90

((Reference)) How many (2a+5)'s (2a+5)'s are there in 3x(2a+5) 3x(2a+5) ?

3x 3x

Exercise 91

((Reference)) Simplify 3(5n+6 m 2 )2(3n+4 m 2 ) 3(5n+6 m 2 )2(3n+4 m 2 ) .

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