1.
Example 1
Factor out a common binomial
Factor the expression.
a. 2x ( x + 4 ) – 3( x + 4 )
b. 3y2 ( y – 2 ) + 5( 2 – y )
SOLUTION
a. 2x ( x + 4 ) – 3( x + 4 ) = ( 2x – 3 ) ( x + 4 ) Distributive
property
b. The binomials y – 2 and 2 – y are opposites. Factor –1
from 2 – y to obtain y – 2 as a common binomial
factor.
2.
Example 1
Factor out a common binomial
3y2 ( y – 2 ) + 5( 2 – y ) = 3y2 ( y – 2 ) – 5( y – 2) Factor – 1 from
(2 – y).
= ( 3y2 – 5 ) ( y – 2)
Distributive
property
3.
Factor by Grouping
In a polynomial with 4 terms, factor a common
monomial from pairs of terms, then look for a
common binomial factor.
4.
Example 2
Factor by grouping
Factor x3 + 3x2 + x + 3.
x3 + 3x2 + x + 3 = ( x3 + 3x2 ) + ( x + 3)
Group terms.
= x2 ( x + 3 ) + 1( x + 3) Factor each
group; write x + 3
as 1(x + 3).
= ( x2 + 1 ) ( x + 3)
Distributive
property
5.
Example 3
Factor by grouping
Factor x3 – 6 + 2x – 3x2
SOLUTION
The terms x3 and – 6 have no common factor. Use the
commutative property to rearrange the terms so that
you can group terms with a common factor.
x3 – 6 + 2x – 3x2 = x3 – 3x2 + 2x – 6
Rearrange terms.
= ( x3 – 3x2 ) + ( 2x – 6) Group terms.
= x2 ( x – 3 ) + 2( x – 3) Factor each group.
= ( x2 + 2 ) ( x – 3)
Distributive
property
6.
Example 3
Factor by grouping
CHECK
Check your factorization using a graphing calculator.
Graph y1 = x3 – 6 + 2x – 3x2 and y2 = ( x – 3) ( x2 + 2 ).
Because the graphs coincide, you know that your
factorization is correct.
7.
Factoring Completely
A factorable polynomial with integer coefficients is
factored completely if it is written as a product of
unfactorable polynomials with integer coefficients.
8.
Guidelines for Factoring Polynomials Completely
1. Factor out the greatest common monomial factor. (9.5)
3x 2 + 6x = 3x(x + 2)
2. Look for a difference of two squares or a perfect square
trinomial. (9.8)
x 2 + 4x + 4 = (x + 2)2
x - 9 = (x + 3)(x - 3)
3. Factor a trinomial of the form ax 2 + bx + c into a
2
product of binomial factors. (9.6 & 9.7)
3x - 5x - 2 = (3x +1)(x - 2)
2
4. Factor a polynomial with four terms by grouping. (9.9)
x3 + x - 4x 2 - 4 = (x - 4)(x 2 +1)
9.
Example 4
Multiple Choice Practice
Which is the completely factored form of 12n2 + 10n – 8?
2(3n – 4 ) (2n + 1 )
2(3n + 4 ) (2n – 1 )
3(3n – 2 ) (2n + 2 )
3(6n – 2 ) ( n + 4 )
SOLUTION
12n2 + 10n – 8 = 2(6n2 + 5n – 4 )
= 2(3n + 4 ) (2n – 1 )
ANSWER
The correct answer is B.
Factor out 2.
Factor trinomial.
10.
Example 5
Solve
Solve a polynomial equation
3x3 + 18x2
= – 24x.
SOLUTION
3x3 + 18x2
3x3 + 18x2
3x ( x2 +
+ 24x
= – 24x
Write original equation.
= 0
Add 24x to each side.
6x + 8 ) = 0
Factor out 3x.
3x ( x + 2 ) ( x + 4 ) = 0
3x = 0 or
x = 0 or
x + 2 = 0
or
x = – 2 or
Factor trinomial.
x + 4 = 0
x = –4
Zero-product property
Solve for x.
11.
Example 5
Solve a polynomial equation
ANSWER
The solutions of the equation are 0, – 2, and – 4.
CHECK
Check each solution by substituting it for x in the
equation. One check is shown here.
?
( – 2 )3 + 18( – 2 )2 = – 24 ( – 2 )
3
?
– 24 + 72 = 48
48 = 48
12.
Example 6
Solve a multi-step problem
TERRARIUM
A large terrarium is used to display a box turtle in a
pet store. The terrarium has the shape of a rectangular
prism with a volume of 8748 cubic inches. The
dimensions of the terrarium are shown. Find the length,
width, and height of the terrarium.
13.
Example 6
Solve a multi-step problem
SOLUTION
STEP 1 Write a verbal model. Then write an equation.
8748
= ( w + 36 )
•
w
•
(w – 9)
STEP 2 Solve the equation for w.
8748 = (w + 36 ) ( w) ( w – 9 )
Write equation.
0 = w3 + 27w2 – 324w – 8748 Multiply. Subtract
8748 from each side.
14.
Example 6
Solve a multi-step problem
0 = (w3 + 27w2 ) – (324w + 8748 )
Group terms.
0 = w2( w + 27 ) – 324( w + 27 )
Factor each group.
0 = (w2 – 324 ) ( w + 27 )
Distributive
property
0 = ( w + 18 ) ( w – 18 ) ( w + 27 )
Difference of two
squares pattern
w + 18 = 0
or w – 18 = 0
w = –18 or w = 18
or w + 27 = 0 Zero-product
or w = –27
property
Solve for w.
Because the width cannot be negative, the only
solution is w = 18
15.
Example 6
Solve a multi-step problem
STEP 3 Find the length and height.
Length w + 36 = 18 + 36 = 54
Height = w – 9 = 18 – 9 = 9
ANSWER
The length is 54 inches, the width is 18 inches, and the
height is 9 inches.
16.
9.9 Warm-Up (Day 1)
Factor the expression.
1.
x(x - 2)+ (x - 2)
2.
a + 3a + a + 3
3.
y + 2x + yx + 2y
3
2
2
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