Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.1 – Slide 1.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.1 – Slide 1


Exponents and Polynomials

Chapter 5


5.1

Adding and Subtracting Polynomials


5.1 Adding and Subtracting Polynomials

Objectives

1. Review combining like terms.2. Know the vocabulary for polynomials.3. Evaluate polynomials.4. Add polynomials.5. Subtract polynomials.6. Add and subtract polynomials with more than

one variable.



Review Combining Like Terms

(a) – 3 x + 7 x 5 5

= (– 3 + 7) x 5

= 4 x 5

= (2 + 5 – 1) n 3

= 6 n 3

(b) 2 n + 5 n – n 3 3 3

Distributive property

Example 1 Simplify each expression by adding like terms.




Review Combining Like Terms

(c) 4 g h – 9 g h + 2 g h 7 7 7 4 4 4

= (4 – 9 + 2) g h 7 4

= – 3 g h 7 4

Example 1 (concluded)Simplify each expression by adding like terms.


Example 2 Simplify each polynomial if possible. Then give the degree and tell whether the polynomial is a monomial, a binomial, a trinomial, or none of these.



Knowing the Vocabulary for Polynomials

(a) 2 t + 7 4 The polynomial cannot be simplified.The degree is 4.The polynomial is a binomial.

The polynomial can be simplified.

The degree is 2.The simplified polynomial is

a monomial.

(b) 3 e + 5 e – 9 e2 2 2

= – e 2

Two terms.

One term.



Evaluating a Polynomial

Substitute h = –3.

Example (a) Find the value of 5h4 – 3h2 + 4h + 7 when h = – 3.

Apply the exponents.

Multiply.

Add and subtract.

4 25 3 4 7h h h

4 23 3 4 35 3 7

5 81 3 9 4 3 7

405 27 12 7

373




Substitute h = 2.

Example (concluded) (b) Find the value of 5h4 – 3h2 + 4h + 7 when h = 2.

Apply the exponents.

Multiply.

Add and subtract.

4 25 3 4 7h h h

4 22 25 3 4 72

5 16 3 4 4 2 7

80 12 8 7

83




CAUTIONUse parentheses around the numbers that are substitutedfor the variable in Example 3, particularly when substitutinga negative number for a variable that is raised to a power. Otherwise, a sign error may result.



Adding Polynomials

Adding PolynomialsTo add two polynomials, add like terms.



Adding Polynomials

Example 4

8y – 7y – y + 33 2

6y + 2y – 4y + 13 2

+ 4– 5y 14y 3 2 – 5y

Write like terms in columns.

Now add, column by column.

3 2 3 2(a) Add 8 7 3 and 6 2 4 1.y y y y y y



Adding Polynomials

+ 12 – 7n 7n 3 2+ n

Write like terms in columns.

Now add, column by column.

n – 9n + 122

7n 3 + 2n

Example 4 (concluded)3 2(b) Add 7 2 and 9 12.n n n n



Adding Polynomials

– 3n 4 – 15n 3 + 6=

Example 54 3 4 3(a) Add 2 7 4 and 5 8 10.n n n n

( 2n – 7n – 4 ) + ( – 5n – 8n + 10 )4 3 4 3



Adding Polynomials

7p 3 – 4p – 5=

3

Example 5 (concluded)3(b) Add 7 9 4 and 5 1.p p p

( 7p – 9p – 4 ) + ( 5p – 1 )



Subtracting Polynomials

Subtracting PolynomialsTo subtract two polynomials, change all the signs of the second polynomial and add the result to the first polynomial.




( 3x – 5 ) – ( 6x – 4 )

Change the signs in the second polynomial and add.

– 3x = – 1

Example 6 (a) Perform the subtraction 3 5 6 4 .x x

= ( 3x – 5 ) + ( – 6x + 4 )




Change the signs in the second polynomial and add.

=

( 7y + 8 ) – ( y + 4y – 2 )3 3 2

6y 3 – 4y 2 + 10

Write the problem.

Example 6 (concluded)3 2 3(b) Subtract 4 2 from 7 8.y y y

= ( 7y + 8 ) + ( – y – 4y + 2 )3 3 2




Arrange like terms in columns..

4g + 6g – g – 54 3

– 6g + 2g – 4g + 34 3 6g – 2g + 4g – 34 3

4g + 6g – g – 54 3

Change all signs in the second row, then add.

+ 4g 3 + 3g – 810g 4

Example 7

4 3 4 3

(b) Subtract by columns:

4g 6 5 6g 2 4 3 .g g g g




Example 8 Perform the indicated operations to simplify the expression.

Rewrite, changing the subtraction to adding the opposite.

Combine like terms.

5= – 8m 2+ 15m

( 2 – 3m + 5m ) + ( – 6 – 4m + 7m ) + ( 9 – m + 3m )2 2 2 =

( 2 – 3m + 5m ) – ( 6 + 4m – 7m ) + ( 9 – m + 3m ).2 2 2



Adding and Subtracting Multivariable Polynomials

Example 9

(a) Add or subtract as indicated.

= 9c + 7cd – 3d

( 6c – 2cd + d ) + ( 3c + 9cd – 4d )



Adding and Subtracting Multivariable Polynomials

Example 9 (concluded)

(b) Add or subtract as indicated.

– ab

( 2a b – 4ab + b ) – ( 5a b – 3ab + 7b )2 2 2 2

2 2 2 2

= – 3a b2 – 6b 2

= 2a b – 4ab + b – 5a b + 3ab – 7b

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