8-Multiplying Polynomials. Mathematics 8

Multiplying Polynomials

1. Chapter 5 Section 5

2. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiply a monomial and a polynomial.
Multiply two polynomials.
Multiply binomials by the FOIL method.
5.5
2
3
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Multiplying Polynomials

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Multiply a monomial and a
polynomial.
Slide 5.5-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
To find the product of a monomial and a polynomial with more than one
term we use the distributive property and multiplication of monomials.
Multiply a monomial and a polynomial.
As shown in Section 5.1, we find the product of two monomials by
using the rules for exponents and the commutative and associative
properties. For example

6 6 6 6 6 6
8 9 8 9 72 .m n m n m n    
Do not confuse addition of terms with multiplication of terms. For
instance,
but
5 5 5
7 2 9q q q  
5 5 5 5 10
7 2 7 2 14 .q q q q

  
Slide 5.5-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
Find the product.
 
4 2
2 3 2 5x x x 
   
424 4
3 2 52 2 2x x xx x     
6 5 4
6 4 10x x x  
Slide 5.5-5
EXAMPLE 1Multiplying Monomials and Polynomials

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Multiply two polynomials.
Slide 5.5-6

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply two polynomials.
We can use the distributive property repeatedly to find the product of
any two polynomials. For example, to find the product of the
polynomials x
2
+ 3x +5 and x − 4, think of x − 4 as a single quantity
and use the distributive property as follows.
 
2 2
4 4 45 5 43 3x x xx x xx x      
Now use the distributive property three more times to find x
2
(x − 4),
3x(x − 4), and 5(x − 4).
Multiplying Polynomials
To multiply two polynomials, multiply each term of the second
polynomial by each term of the first polynomial and add the products.

2 2
3 3 5 54 4 4x xx x x x x       
  
3 2 2
4 3 12 5 20x x x x x        
3 2
7 20x x x   
Slide 5.5-7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply (m
3
− 2m + 1)·(2m
2
+ 4m + 3).
Solution:


3 2 3 3 2
2
2 4 3 2 2 2 4
2 3 1 2 1 4 1 3
m m m m m m m m m
m m m
      
    
  
5 4 3 3 2 2
2 4 3 4 8 6 2 4 3m m m m m m m m           
5 4 3 2
2 4 6 2 3m m m m m     
Slide 5.5-8
EXAMPLE 2Multiplying Two Polynomials

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply.
2
3 4 5x x 
4x
2
12 16 20x x 
3 2
3 4 5x x x 
3 2
3 16 11 20x x x  
Solution:
Slide 5.5-9
EXAMPLE 3Multiplying Polynomials Vertically

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply.
3 2
5 10 20x x 
21 2
5 5
x
3 2
2 4 8x x 
5 4 3 2
2 0 4x x x x  
5 4 3
2 2 8x x x  
Solution:
Slide 5.5-10
EXAMPLE 4
Multiplying Polynomials with Fractional Coefficients Vertically

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Multiply binomials by the FOIL
method.
Slide 5.5-11

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply binomials by the FOIL method.
In algebra, many times the polynomials to be multiplied are
binomials. For these products, the FOIL method reduces the
rectangle method to a systematic approach without the rectangle.
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 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.
3 5x x 
2
Fx L15
O5x I3x
Slide 5.5-12

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the FOIL to find the product.
2 6x x 
2
6 2 8x x x   
2
8 12x x  
Solution:
2
Fx
O 6xI 2x
L 12
Slide 5.5-13
EXAMPLE 5Using the FOIL Method

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Multiply  5 6 2 3 .x y 
10 15 12 18xy x y   
Solution:
  5 6 2 3x y 
F 10xy L 18
O 15x I 12y
Slide 5.5-14
EXAMPLE 6Using the FOIL Method

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Find each product.
  4 2 3y x y x  
2 2
8 12 2 3y xy xy x   
Solution:
 
3
3 2 2 1x x x 
2 2
8 14 3y xy x  
 
3 2
3 2 1 4 2x x x x   
 
3 2
3 2 3 2x x x  
5 4 3
6 9 6x x x  
Slide 5.5-15
EXAMPLE 7Using the FOIL Method