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4.5: Combining Polynomials Using Addition and Subtraction

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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 4.5.1

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

Example 4.5.2

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 4.5.3

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

Example 4.5.4

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

6x+4x+2y=10x+2y

Practice Set A

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

Practice Problem 4.5.1

Like terms with different numerical coefficients represent

Answer

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.

Simplify each of the following polynomials by combining like terms.

Example 4.5.5

2x+5x+3x.

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

Example 4.5.6

7x+8y3x.

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

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

Example 4.5.7

4a32a2+8a3+a22a3.

4a3,8a3, and 2a3 represent quantities of the same type.

4a3+8a32a3=10a3.

2a2 and a2 represent quantities of the same type.

2a2+a2=a2.

Thus,

4a32a2+8a3+a22a3=10a3a2.

Practice Set B

Simplify each of the following expressions.

Practice Problem 4.5.2

4y+7y

Answer

11y

Practice Problem 4.5.3

3x+6x+11x

Answer

20x

Practice Problem 4.5.4

5a+2b+4ab7b

Answer

9a6b

Practice Problem 4.5.5

10x34x3+3x212x3+5x2+2x+x3+8x

Answer

5x3+8x2+10x

Practice Problem 4.5.6

2a5a5+14ab9+9ab23a5.

Answer

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.

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 4.5.8

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 4.5.9

4x+9(x26x2)+5 Remove parentheses.

4x+9x254x18+5 Combine like terms.

50x+9x213

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.

9x250x13

Example 4.5.10

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

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

Example 4.5.11

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

x23x+12x2+18x Combine like terms.

13x2+15x

Practice Set C

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

Practice Problem 4.5.7

4(x+6)+3(2+x+3x2)2x2

Answer

7x2+7x+30

Practice Problem 4.5.7

7(x+x3)4x3x+1+4(x22x3+7)

Answer

5x3+4x2+6x+29

Practice Problem 4.5.7

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

Answer

59a+27

Practice Problem 4.5.7

x(x+3)+4x2+2x

Answer

5x2+5x

Practice Problem 4.5.7

a3(a2+a+5)+a(a4+3a2+4)+1

Answer

2a5+a4+8a3+4a+1

Practice Problem 4.5.7

2[83(x3)]

Answer

6x+34

Practice Problem 4.5.7

x2+3x+7[x+4x2+3(x+x2)]

Answer

50x2+31x

Exercises

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

Exercise 4.5.1

x+3x

Answer

4x

Exercise 4.5.2

4x+7x

Exercise 4.5.3

9a+12a

Answer

21a

Exercise 4.5.4

5m3m

Exercise 4.5.5

10x7x

Answer

3x

Exercise 4.5.6

7y9y

Exercise 4.5.7

6k11k

Answer

5k

Exercise 4.5.8

3a+5a+2a

Exercise 4.5.9

9y+10y+2y

Answer

21y

Exercise 4.5.10

5m7m2m

Exercise 4.5.11

h3h5h

Answer

7h

Exercise 4.5.12

a+8a+3a

Exercise 4.5.13

7ab+4ab

Answer

11ab

Exercise 4.5.14

8ax+2ax+6ax

Exercise 4.5.15

3a2+6a2+2a2

Answer

11a2

Exercise 4.5.16

14a2b+4a2b+19a2b

Exercise 4.5.17

10y15y

Answer

5y

Exercise 4.5.18

7ab9ab+4ab

Exercise 4.5.19

210ab4+412ab4+100a4b (Look closely at the exponents.)

Answer

622ab4+100a4b

Exercise 4.5.20

5x2y0+3x2y+2x2y+1,y0 (Look closely at the exponents.)

Exercise 4.5.21

8w212w23w2

Answer

7w2

Exercise 4.5.22

6xy3xy+7xy18xy

Exercise 4.5.23

7x32x210x+15x23x312+x

Answer

4x37x29x11

Exercise 4.5.24

21y15x+40xy611y+712xxy

Exercise 4.5.25

1x+1y1x1y+xy

Answer

xy

Exercise 4.5.26

5x23x7+2x2x

Exercise 4.5.27

2z+15z+4z3+z26z2+z

Answer

2z35z2+16z

Exercise 4.5.28

18x2y14x2y20x2y

Exercise 4.5.29

9w59w49w5+10w4

Answer

18w5+w4

Exercise 4.5.30

2x4+4x38x2+12x17x31x46x+2

Exercise 4.5.31

17d3r+3d3r5d3r+6d2r+d3r30d2r+37+2

Answer

16d3r24d2r2

Exercise 4.5.32

a0+2a04a0,a0

Exercise 4.5.33

4x0+3x05x0+7x0x0,x0

Answer

8

Exercise 4.5.34

2a3b2c+3a2b2c0+4a2b2a3b2c,c0

Exercise 4.5.35

3z6z+8z

Answer

5z

Exercise 4.5.36

3z2z+3z3

Exercise 4.5.37

6x3+12x+5

Answer

6x3+12x+5

Exercise 4.5.38

3(x+5)+2x

Exercise 4.5.39

7(a+2)+4

Answer

7a+18

Exercise 4.5.40

y+5(y+6)

Exercise 4.5.41

2b+6(35b)

Answer

28b+18

Exercise 4.5.42

5a7c+3(ac)

Exercise 4.5.43

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

Answer

31x+8

Exercise 4.5.44

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

Exercise 4.5.45

(a+5)4+6a20

Answer

10a

Exercise 4.5.46

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

Exercise 4.5.47

(10x+3y2)4+4(10x+3y2)

Answer

80x+24y2

Exercise 4.5.48

2(x6)+5

Exercise 4.5.49

1(3x+15)+2x12

Answer

5x+3

Exercise 4.5.50

1(2+9a+4a2)+a211a

Exercise 4.5.51

1(2x6b+6a2b+8b2)+1(5x+2b3a2b

Answer

3a2b+8b24b+7x

Exercise 4.5.52

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

Exercise 4.5.53

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

Answer

yes

Exercise 4.5.54

3(2a+2a2)+8(3a+3a2)

Exercise 4.5.55

x(x+2)+2(x2+3x4

Answer

3x2+8x8

Exercise 4.5.56

A(A+7)+4(A2+3a+1)

Exercise 4.5.57

b(2b3+5b2+b+6)6b24b+2

Answer

2b4+5b35b2+2b+2

Exercise 4.5.58

4aa(a+5)

Exercise 4.5.59

x3x(x27x1)

Answer

3x3+21x2+4x

Exercise 4.5.60

ab(a5)4a2b+2ab2

Exercise 4.5.61

xy(3xy+2x5y)2x2y25x2y+4xy2

Answer

x2y23x2yxy2

Exercise 4.5.62

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

Exercise 4.5.63

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

Answer

52k2+6k

Exercise 4.5.64

8a[2a4ab+9(a5ab)]

Exercise 4.5.65

6m+5n[n+3(n1)]+2n24n29m

Answer

128n290n3m

Exercise 4.5.66

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

Exercise 4.5.67

89[b2a+6c(c+4)4c2]+4a+b3b

Answer

144c2112a+77b+1728c

Exercise 4.5.68

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

Exercise 4.5.69

3xy2(4xy+5y)+2xy3+6x2y3+4y312xy3

Answer

18x2y3+5xy3+4y3

Exercise 4.5.70

9a3b7(a3b52a2b2+6)2a(a2b75a5b12+3a4b9)a3b7

Exercise 4.5.71

8(3a+2)

Answer

24a16

Exercise 4.5.72

4(2x3y)

Exercise 4.5.73

4xy2[7xy6(5xy2)+3(xy+1)+1]

Answer

24x2y416x2y3+104xy2

Exercises for Review

Exercise 4.5.74

Simplify (x10y8z2x2y6)3

Exercise 4.5.75

Find the value of 3(49)6(3)123

Answer

4

Exercise 4.5.76

Write the expression 42x2y5z321x4y7 so that no denominator appears.

Exercise 4.5.77

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

Answer

3x

Exercise 4.5.78

Simplify 3(5n+6m2)2(3n+4m2)


This page titled 4.5: Combining Polynomials Using Addition and Subtraction is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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