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Algebraic Expressions and Equations: Combining Polynomials Using Addition and Subtraction

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

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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.

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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

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

Example 2

6 houses+4 houses+2 motels=10 houses+2 motels 6 houses+4 houses+2 motels=10 houses+2 motels . 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+8y−3x 7x+8y−3x .
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+8y−3x=4x+8y 7x+8y−3x=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.

Exercise 2

4y+7y 4y+7y

Solution

11y 11y

Exercise 3

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

Solution

20x 20x

Exercise 4

5a+2b+4a−b−7b 5a+2b+4a−b−7b

Solution

9a−6b 9a−6b

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 +1−4ab−9+9ab−2−3− a 5 2 a 5 − a 5 +1−4ab−9+9ab−2−3− a 5

Solution

5ab−13 5ab−13

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.

Distributive Property

The product of a monomial a and a binomial b plus c is equal to ab plus ac. This is the distributive property. In the expression, there are two arrows originating from the monomial, a, and pointing towards the terms b and c of the binomial.

Sample Set C

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

Example 8

Simplifying the expression six a plus the product of five and the binomial a plus three, using the distributive property, and combining like terms. See the longdesc for a full description.

Example 9

4x+9( x 2 −6x−2)+5 Remove parentheses. 4x+9 x 2 −54x−18+5 Combine like terms. −50x+9 x 2 −13 4x+9( x 2 −6x−2)+5 Remove parentheses. 4x+9 x 2 −54x−18+5 Combine like terms. −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 −50x−13 9 x 2 −50x−13

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] Remove this set of parentheses. 2+18+8a Combine like terms. 20+8a Write in descending order. 8a+20 2+2[9+4a] Remove this set of parentheses. 2+18+8a Combine like terms. 20+8a Write in descending order. 8a+20

Example 11

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

x 2 −3x+12 x 2 +18x Combine like terms. 13 x 2 +15x x 2 −3x+12 x 2 +18x Combine like terms. 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)+6a−7+(8+4) (a+3a+2) 5(a+2)+6a−7+(8+4) (a+3a+2)

Solution

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[8−3(x−3)] 2[8−3(x−3)]

Solution

−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.

Exercise 14

x+3x x+3x

Solution

4x 4x

Exercise 15

4x+7x 4x+7x

Exercise 16

9a+12a 9a+12a

Solution

21a 21a

Exercise 17

5m−3m 5m−3m

Exercise 18

10x−7x 10x−7x

Solution

3x 3x

Exercise 19

7y−9y 7y−9y

Exercise 20

6k−11k 6k−11k

Solution

−5k −5k

Exercise 21

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

Exercise 22

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

Solution

21y 21y

Exercise 23

5m−7m−2m 5m−7m−2m

Exercise 24

h−3h−5h h−3h−5h

Solution

−7h −7h

Exercise 25

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

Exercise 26

7ab+4ab 7ab+4ab

Solution

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

Solution

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

Exercise 30

10y−15y 10y−15y

Solution

−5y −5y

Exercise 31

7ab−9ab+4ab 7ab−9ab+4ab

Exercise 32

210a b 4 +412a b 4 +100 a 4 b (Look closely at the exponents.) 210a b 4 +412a b 4 +100 a 4 b (Look closely at the exponents.)

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, y≠0 (Look closely at the exponents.) 5 x 2 y 0 +3 x 2 y+2 x 2 y+1, y≠0 (Look closely at the exponents.)

Exercise 34

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

Solution

−7 w 2 −7 w 2

Exercise 35

6xy−3xy+7xy−18xy 6xy−3xy+7xy−18xy

Exercise 36

7 x 3 −2 x 2 −10x+1−5 x 2 −3 x 3 −12+x 7 x 3 −2 x 2 −10x+1−5 x 2 −3 x 3 −12+x

Solution

4 x 3 −7 x 2 −9x−11 4 x 3 −7 x 2 −9x−11

Exercise 37

21y−15x+40xy−6−11y+7−12x−xy 21y−15x+40xy−6−11y+7−12x−xy

Exercise 38

1x+1y−1x−1y+x−y 1x+1y−1x−1y+x−y

Solution

x−y x−y

Exercise 39

5 x 2 −3x−7+2 x 2 −x 5 x 2 −3x−7+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 y−14 x 2 y−20 x 2 y 18 x 2 y−14 x 2 y−20 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 +12x−1−7 x 3 −1 x 4 −6x+2 2 x 4 +4 x 3 −8 x 2 +12x−1−7 x 3 −1 x 4 −6x+2

Exercise 44

17 d 3 r+3 d 3 r−5 d 3 r+6 d 2 r+ d 3 r−30 d 2 r+3−7+2 17 d 3 r+3 d 3 r−5 d 3 r+6 d 2 r+ d 3 r−30 d 2 r+3−7+2

Solution

16 d 3 r−24 d 2 r−2 16 d 3 r−24 d 2 r−2

Exercise 45

a 0 +2 a 0 −4 a 0 , a≠0 a 0 +2 a 0 −4 a 0 , a≠0

Exercise 46

4 x 0 +3 x 0 −5 x 0 +7 x 0 − x 0 , x≠0 4 x 0 +3 x 0 −5 x 0 +7 x 0 − x 0 , x≠0

Solution

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, c≠0 2 a 3 b 2 c+3 a 2 b 2 c 0 +4 a 2 b 2 − a 3 b 2 c, c≠0

Exercise 48

3z−6z+8z 3z−6z+8z

Solution

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

Solution

7a+18 7a+18

Exercise 53

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

Exercise 54

2b+6(3−5b) 2b+6(3−5b)

Solution

−28b+18 −28b+18

Exercise 55

5a−7c+3(a−c) 5a−7c+3(a−c)

Exercise 56

8x−3x+4(2x+5)+3(6x−4) 8x−3x+4(2x+5)+3(6x−4)

Solution

31x+8 31x+8

Exercise 57

2z+4ab+5z−ab+12(1−ab−z) 2z+4ab+5z−ab+12(1−ab−z)

Exercise 58

(a+5)4+6a−20 (a+5)4+6a−20

Solution

10a 10a

Exercise 59

(4a+5b−2)3+3(4a+5b−2) (4a+5b−2)3+3(4a+5b−2)

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

Exercise 61

2(x−6)+5 2(x−6)+5

Exercise 62

1(3x+15)+2x−12 1(3x+15)+2x−12

Solution

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(2x−6b+6 a 2 b+8 b 2 )+1(5x+2b−3 a 2 b) 1(2x−6b+6 a 2 b+8 b 2 )+1(5x+2b−3 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?

Solution

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 +3x−4) x(x+2)+2( x 2 +3x−4)

Solution

3 x 2 +8x−8 3 x 2 +8x−8

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

4a−a(a+5) 4a−a(a+5)

Exercise 72

x−3x( x 2 −7x−1) x−3x( x 2 −7x−1)

Solution

−3 x 3 +21 x 2 +4x −3 x 3 +21 x 2 +4x

Exercise 73

ab(a−5)−4 a 2 b+2ab−2 ab(a−5)−4 a 2 b+2ab−2

Exercise 74

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

Solution

x 2 y 2 −3 x 2 y−x y 2 x 2 y 2 −3 x 2 y−x 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[2a−4ab+9(a−5−ab)] 8a[2a−4ab+9(a−5−ab)]

Exercise 78

6{m+5n[n+3(n−1)]+2 n 2 }−4 n 2 −9m 6{m+5n[n+3(n−1)]+2 n 2 }−4 n 2 −9m

Solution

128 n 2 −90n−3m 128 n 2 −90n−3m

Exercise 79

5[4(r−2s)−3r−5s]+12s 5[4(r−2s)−3r−5s]+12s

Exercise 80

8{9[b−2a+6c(c+4)−4 c 2 ]+4a+b}−3b 8{9[b−2a+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(6x−3)+x]−2x−25x+4 5[4(6x−3)+x]−2x−25x+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

Exercise 84

−8(3a+2) −8(3a+2)

Solution

−24a−16 −24a−16

Exercise 85

−4(2x−3y) −4(2x−3y)

Exercise 86

−4x y 2 [7xy−6(5−x y 2 )+3(−xy+1)+1] −4x y 2 [7xy−6(5−x 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(4−9)−6(−3)−1 2 3 −3(4−9)−6(−3)−1 2 3 .

Solution

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) ?

Solution

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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