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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 5.3
Factoring Polynomials
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Common Factors
• Factoring and Equations
• Factoring by Grouping
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Common Factors
When factoring a polynomial, we first look for factors that are common to each term. By applying the distributive property, we can write a polynomial as two factors.For example:
It can be factored as follows: 2 22 4 )2(xx x x
2
2 2
2 4
2
2
24
x
x
x
x x
x
x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Factor. a. b. c. d.
Solutiona. b.
26 7x x 3 215 5x x 3 28 2 4w w w 2 3 2 26x y x y
26 7x x26 6
7 7
(6 7)
x x
x
x
x
x x
3 215 5x x3
2
2
2
2
15 3
5
(3 1)
5
5
5
x x
x
x
x
x
x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Factor. a. b. c. d.
Solutionc. d.
26 7x x 3 215 5x x 3 28 2 4w w w 2 3 2 26x y x y
3 28 2 4w w w 3 2
2
2
8 4
2
4 2
2 (4
2
2
)
2
2
w w
w w
w
w
ww
w w w
2 3 2 26x y x y2 3
2 2
2
2 2
2 2
26 6
(6 1)
x y
x
x y
x y y
x y y
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Factor.a. b.
Solutiona.
b.
6 4 332 16 8x x x 29 3 12m n mn m
6
4
3
3
3
3
3
32
1
48
6
8 8
28
x
x x
x
x
x
x
x
33(4 2 1)8 xx x 6 4 332 16 8x x x
29 3 12m n mn m 2 3
3
3
39
3
2 41
m n
mn
m n
m
m
m
n
m
43 (3 )mnm n
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Factoring and Equations
To solve equations using factoring, we use the zero-product property. It states that, if the product of two numbers is 0, then at least one of the numbers must equal 0.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve each equation.a. b. Solutiona. b.
3 ( 4) 0x x (4 1)(3 4) 0x x
3 ( 4) 0x x 04( )3x x
3 0 or 4 0 x x
0 o r 4 x x
(4 1)(3 4) 0x x 3( 4 01)(4 )xx 0 or 31 044 xx
4 1 or 3 4x x 1 4
or 4 3
x x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve each polynomial equation.a. b. Solution
2 8 0x x 34 4 0x x
a. We begin by factoring out the greatest common factor.
2 8 0
( 8) 0
x x
x x
0 or 8 0x x
0 or 8x x
b. We begin by factoring out the greatest common factor. 3
2
4 4 0
4 ( 1) 0
x x
x x
24 0 or 1 0x x
20 or 1x x
No real number can satisfy x2 = –1, the only solution is 0.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Polynomial equations can also be solved numerically and graphically.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Solve the equation 6x – x2 = 0 numerically, graphically, and symbolically. SolutionNumerical: Make a table of values.
x y
1 7
0 0
1 5
2 8
3 9
4 8
5 5
6 0
Graphical: Plot the points in the table.
The intercepts are the solution to the equation.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Solve the equation 6x – x2 = 0 numerically, graphically, and symbolically. SolutionSymbolic: Start by factoring the left side of the equation.
Note that the numerical and graphical solutions agree with the symbolic solutions.
26 0x x (6 ) 0x x
0 or 6 0x x 0 or 6x x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Factor. a. 3x(x + 1) + 4(x + 1) b. 3x2(2x – 1) – x(2x – 1)Solutiona. Both terms in the expression contain the binomial x + 1. Use the distributive property to factor.
b.
1( ) 4 )1(3 xx x ( )(3 14 ) x x
23 (2 1) (2 1)x x x x 2( ) (2 )1 13 2x xx x 2( )(23 1) x x x
( )(3 )21 1x xx
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Factor the polynomial.
Solution
3 24 3 12x x x
3 24 3 12x x x 3 2( 4 ) (3 12)x x x
3 2( 4 ) (3 12)x x x 2( ) 3( )4 4x x x
2( )( )3 4x x
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Factor the polynomial.
Solution
3 215 10 3 2x x x
3 215 10 3 2x x x 3 2(15 10 ) ( 3 2)x x x
2( )3 2 3 2)5 1( xx x
2( )(35 1 )2x x
