Download Multiplying Polynomials

5.5
1.
2.
3.
4.
Multiplying Polynomials; Special
Products
Multiply a polynomial by a monomial.
Multiply binomials.
Multiply polynomials.
Determine the product when given special polynomial
factors.
Objective 1
Multiply a polynomial by a monomial.
Multiply.
2 p  6 p 2  2 p  1
2p
6 p
2
 2 p  1
2p • 6p2
2p • 2p
2p • –1
2p ∙ 6p2 + 2p ∙ 2p + 2p ∙ – 1
 12 p3  4 p 2  2 p
2 p  6 p  2 p  1  12 p  4 p  2 p
2
3
2
Multiplying a Polynomial by a Monomial
Use the distributive property to multiply each
term in the polynomial by the monomial.
Multiply:

2
5a 2a  a  6

10a3  5a2  30a
When
2a anbequation
ab issolved
5athe answer
 4bc
 3a inbonevariable

is a point on a line
2
3
3
4
.
 6a 5b2  2a3b4  10a6 b  8a2b2c
2 4
1 3 1 2
1

5r  r 
r  r 
r  1
20
5
10


1 5
1 3
4
5r  r  r  r  5r 2
4
2
6
Objective 2
Multiply binomials.
Multiply.
 x  7  x  3
x•3
 x  7
x•x
 x  3
7•x
7•3
 x  x  x 3 7 x  73
= x 2  3x  7 x  21
= x 2  10 x  21
 x  7  x  3 = x 2  10 x  21
Multiplying Polynomials
1. Multiply every term in the second polynomial by
every term in the first polynomial.
2. Combine like terms.
Multiply.
 2x  1 x  5
FOIL: First Outside Inside Last
 2x  1
2x • (–5)
Outside
2x • x
First
1•x
Inside
 x  5
1 • (–5)
Last
Last
First
Outside Inside
 2x  x  2 x   5  1  x  1   5
 2 x 2  10 x  x
 2x2
 9x
 2 x2  9 x  5
5
5
Multiply:
a  4a  9
2
a  13a  36
n  7n  5
n2  2n  35
When an equation in one variable is solved the answer is a point on a line.
3x  5 2 x  7 
2
6x  11x  35
2t  73t  1
6t 2  23t  7
The product of two binomials can be
shown in terms of geometry.
x
7
x
x2
7x
5
5x
35
Length • width = Sum of the areas of the
four internal rectangles
 x  7  x  5
 x 2  5 x  7 x  35
 x 2  12 x  35
Combine like
terms.
Objective 3
Multiply polynomials.
Multiply.  x  3  2 x 2  3x  3
x • 3x
x • 2x2
 x  3
Horizontal Multiplication
x•3
2
2
x
  3 x  3
(–3) • 2x2
(–3) • 3x
(–3) • 3
2
 x  2x2  x  3x  x  3   3  2 x   3  3x   3  3
 2x3
 3x 2
 3x
 2x3  3x 2  6 x  9
 6 x2
 9x
9
Multiply.  x  3  2 x 2  3x  3
Vertical Multiplication
2 x 2  3x  3
x3
2
 6x  9x  9
2
3
 2x  3x  3x
2x
3
 3x  6x  9
2
Multiply:
x  3x2
 3x  2

x 3  3x 2  2 x
 3x 2  9 x  6
x3
 7x  6
When an equation in one variable is solved the answer is a point on a line.
2f

 3g 2f  3fg  9g
2
2

3
2
4 f  27 fg  27g
3
Objective 4
Determine the product when given
special polynomial factors.
Multiply:
a  4a  4
2
a  16
n  5n  5
n2  25
When an equation in one variable is solved the answer is a point on a line.
2t  72t  7
4t 2  49
4b  5c 4b  5c 
16b2  25c 2
Conjugates: Binomials that differ only in the sign
separating the terms.
x + 9 and x – 9
2x + 3 and 2x – 3
–6x + 5 and –6x – 5
Multiplying Conjugates
If a and b are real numbers, variables, or
expressions, then (a + b)(a – b) = a2 – b2.
Multiply:
a  4a  4
n  5n  5
a2  8a  16
n2  10n  25
2 ∙ 4a
2∙–5n
2

a  4
nis apoint5on a line
When an equation in one variable is solved the answer
2
.
2t  72t  7
4b  5c 4b  5c 
4t 2  28t  49
16b2  40bc  25c 2
2t  72
4b  5c 2
Squaring a Binomial
If a and b are real numbers, variables, or
expressions, then
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Remember the shortcut
or
rewrite and foil !!
Multiply:
9k  22
y  52
y 2  10y  25
81k 2  36k  4
When an equation in one variable is solved the answer is a point on a line.
4 t  1
2
16t 2  8t  1
6  7c 
2
36  84c  49c 2
Multiply.  2 y  5 y  3
a) 2 y 2  y  15
b) 2 y  2 y  15
2
c) 2 y 2  y  15
d) 2 y 2  11y  15
5.5
Copyright © 2011 Pearson Education, Inc.
Slide 5- 22
Multiply.  2 y  5 y  3
a) 2 y 2  y  15
b) 2 y  2 y  15
2
c) 2 y 2  y  15
d) 2 y 2  11y  15
5.5
Copyright © 2011 Pearson Education, Inc.
Slide 5- 23
Multiply.  5 x  4 
2
a) 25 x 2  40 x  8
b) 25 x 2  20 x  8
c) 25 x 2  20 x  16
d) 25 x 2  40 x  16
5.5
Copyright © 2011 Pearson Education, Inc.
Slide 5- 24
Multiply.  5 x  4 
2
a) 25 x 2  40 x  8
b) 25 x 2  20 x  8
c) 25 x 2  20 x  16
d) 25 x 2  40 x  16
5.5
Copyright © 2011 Pearson Education, Inc.
Slide 5- 25