Section 12.4 Adding and Subtracting Polynomials

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2
Exponents and
Polynomials
12

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1. Identify terms and coefficients.
2. Combine like terms.
3. Know the vocabulary for polynomials.
4. Evaluate polynomials.
5. Add polynomials.
6. Subtract polynomials.
7. Add and subtract polynomials with more
than one variable.
Objectives
12.4 Adding and Subtracting Polynomials

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Combining Like Terms
(a) –3 x + 7 x5 5
= (–3 + 7) x5
= 4 x5
= (2 + 5 – 1) n 3
= 6 n 3
(b) 2 n + 5 n – n3 3 3
Distributive property
Example
Simplify each expression by adding like terms.
Distributive property

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Combining Like Terms
(c) 4 g h – 9 g h + 2 g h7 7 74 4 4
= (4 – 9 + 2) g h7 4
= – 3 g h7 4
Example (cont)
Simplify each expression by adding like terms.
Distributive property

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example 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.
Classifying Polynomials
(a) 2 t + 74 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.

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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 2
5 3 4 7h h h  
     
4 2
3 3 4 35 3 7    
     5 81 3 9 4 3 7    
405 27 12 7   
373

7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Evaluating a Polynomial
Substitute h = 2.
Example (cont)
(b) Find the value of 5h4 – 3h2 + 4h + 7 when h = 2.
Apply the exponents.
Multiply.
Add and subtract.
4 2
5 3 4 7h h h  
     
4 2
2 25 3 4 72   
     5 16 3 4 4 2 7   
80 12 8 7   
83

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Add Polynomials
Adding Polynomials
To add two polynomials, combine (add) like terms.

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Add Polynomials
Example
8y – 7y – y + 33 2
6y + 2y – 4y + 13 2
+ 4– 5y14y 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     

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add Polynomials
+ 12– 7n7n 3 2+ n
Write like terms in columns.
Now add, column by column.
n – 9n + 122
7n 3 + 2n
Example (cont)
3 2
(b) Add 7 2 and 9 12.n n n n  

11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add Polynomials
– 3n4 – 15n 3
+ 6=
Example
4 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

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Add Polynomials
7p 3 – 4p – 5=
3
Example (cont)
3
(b) Add 7 9 4 and 5 1.p p p  
( 7p – 9p – 4 ) + ( 5p – 1 )

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Polynomials
Subtracting Polynomials
To subtract two polynomials, change all the signs of the
second polynomial and add the result to the minuend (first
polynomial).

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Subtract Polynomials
( 3x – 5 ) – ( 6x – 4 )
Change the signs in the second polynomial and add.
– 3x= – 1
Example
   (a) Perform the subtraction 3 5 6 4 .x x  
= ( 3x – 5 ) + ( – 6x + 4 )

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Subtract Polynomials
Change the signs in the second polynomial and add.
=
( 7y + 8 ) – ( y + 4y – 2 )3 3 2
6y 3
– 4y2
+ 10
Write the problem.
Example (cont)
3 2 3
(b) Subtract 4 2 from 7 8.y y y  
= ( 7y + 8 ) + ( – y – 4y + 2 )3 3 2

16. Slide - 16Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Polynomials
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.
+ 4g3
+ 3g – 810g 4
Example
   4 3 4 3
Subtract by columns:
4g 6 5 6g 2 4 3 .       g g g g

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Subtract Polynomials
Example 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

18. Slide - 18Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add and Subtract Multivariable Polynomials
Example
(a) Add or subtract as indicated.
= 9c + 7cd – 3d
( 6c – 2cd + d ) + ( 3c + 9cd – 4d )

19. Slide - 19Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add and Subtract Multivariable Polynomials
Example (cont)
(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