Section 12.5 Multiplying Polynomials

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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Exponents and
Polynomials
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2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Multiply monomials.
2. Multiply a monomial and a polynomial.
3. Multiply two polynomials.
4. Multiply binomials by the FOIL method.
Objectives
12.5 Multiplying Polynomials

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply a Monomial and a Polynomial
(a) 5x ( 6x + 7 )2 4
Distributive property= 5x ( 6x )2 4 + 5x ( 7 )2
= 30x + 35x6 2 Multiply monomials.
Example Find each product.

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply a Monomial and a Polynomial
Distributive property
= – 2h (– 3h )4 9
(b) – 2h ( – 3h + 8h – 1 )4 9 2
+ (– 2h ) (8h )4 2 + (– 2h )(– 1)4
Example (cont) Find each product.
Multiply monomials.= 6h 13 – 16h 6 + 2h 4

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Two Polynomials
Multiplying Polynomials
To multiply two polynomials, multiply each term of the
second polynomial by each term of the first polynomial
and add the products.

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Two Polynomials
Distributive property( 2y – 5 ) ( 2y – 7y + 4 )2 3
= (2y )2 (2y )3 (–7y)(2y )2+ (4)(2y )2+
(–7y)+ (–5) (4)+ (–5)(–5)(2y )3+
= 4y 5 14y 3– 8y 2+ – 20+ 35y– 10y 3
= 4y 5 24y 3– 8y 2+ + 35y – 20 Combine like terms.
  2 3
Multiply 2 5 2 7 4 .y y y  Example

7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
5n + 8n – 3n – 23 2
Multiply Two Polynomials
4n + 32
– 6– 9n+ 24n 215n 3
– 8n 2– 12n 3+ 32n 420n 5
– 6– 9n+ 3n 3+ 32n 420n 5 + 16n 2
  2 3 2
Multiply 4 3 5 8 3 2 vertically.n n n n   Example

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Binomials by the FOIL Method
Multiplying Binomials by the FOIL Method
Step 1 Multiply the two First terms of the binomials to
get the first term of the answer.
Step 2 Find the Outer product and the Inner product
and add them (when possible) to get the
middle term of the answer.
Step 3 Multiply the two Last terms of the binomials to
get the last term of the answer.

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Binomials by the FOIL Method
F
( y – 4 ) ( 3y – 2 )
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y ( 3y )
y ( – 2 )
– 4 ( 3y )
– 4 (– 2 )
= 3y – 2y – 12y + 82
= 3y – 14y + 82
F O I L
Example   Use the FOIL method to find 4 3 2 .y y 
Multiply the last terms:
Multiply the inner terms:
Multiply the outer terms:
Multiply the first terms:

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Binomials by the FOIL Method
F
( n – 7 ) ( 5n + 1 )
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n ( 5n )Multiply the first terms:
n ( 1 )Multiply the outer terms:
– 7 ( 5n )Multiply the inner terms:
– 7 ( 1 )Multiply the last terms:
= 5n + n – 35n – 72
= 5n – 34n – 72
F O I L
Example   Multipy 7 5 1 .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
Multiply Binomials by the FOIL Method
F
( 3g + 2 ) ( 9g – 4 )
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3g ( 9g )Multiply the first terms:
3g ( – 4 )Multiply the outer terms:
2 ( 9g )Multiply the inner terms:
2 ( – 4 )Multiply the last terms:
= 27g – 12g + 18g – 82
= 27g + 6g – 82
F O I L
Example   Find the product 3 2 9 4 . g g

12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Multiply Binomials by the FOIL Method
F
( 6a + 3b ) ( 4a – 2b )
O
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L
6a ( 4a )Multiply the first terms:
6a ( – 2b )Multiply the outer terms:
3b ( 4a )Multiply the inner terms:
3b ( – 2b )Multiply the last terms:
F O I L
= 24a – 12ab + 12ab – 6b2 2
= 24a – 6b2 2
Example   Find the product 6 3 4 2 .a b a b 