7.8 Summary of
Factoring
CORD Math
Mrs. Spitz
Fall 2006
Objective:
• Factor polynomials by applying various
methods of factoring.
Assignment
• Pg. 288 #5-50 all
Introduction
• In this chapter, you have used various
methods to factor different types of
polynomials. The following chart on the
next slide summarizes these methods
and can help you decide when to use a
specific method.
Summary Chart
Check for:
Greatest
Common Factor
Difference of
squares
Number of Terms
Two
Three
Four +
X
X
X
X
X
Perfect Square
trinomial
X
Trinomial w/ two
binomial factors
X
Pairs of terms w/
common
monomial factor
X
Watch . . .
• Whenever there is a GCF other than 1,
always factor it out first. Then check the
appropriate factoring methods in the
order shown in the table. Use these
methods to factor until all of the factors
are prime.
Ex. 1: Factor
2
3x
- 27
• First check for a GCF. Then since there
are two terms, check for the difference
of squares.
3x  27
2
 3( x  9)
2
 3( x  3)( x  3)
3 is the GCF
x2 – 9 is the difference of
squares since x  x = x2
and 3  3 = 9
• Thus 3x2 – 27 is completely factored as
3(x +3)(x – 3)
Ex. 2: Factor
•
2
9y
– 58y + 49
The polynomial has three terms. So
check for the following:
1. GCF – The GCF is 1
2. Perfect square trinomial – Although
9y2 = (3y)2 and 49 = (7)2, 58y ≠
2(3y)(7).
3. Trinomial w/ two binomial factors: Are
there two numbers whose product is 9
 49 or 441 and whose sum is -58?
Ex. 2: Factor 9y2 – 58y + 49
• You must find two numbers whose product is
9 · 49 or 441 and whose sum is -58.
Factors of 441
-1, -441
-3, -147
-7, -63
-9, -49
-21, -21
Sum of factors
-1 + -441=-442
-3 + -147=-150
-7 + -63 = -70
-9 + -49 = -58
-21 + -21 = -42
Ex. 2: Factor
•
•
2
9y
– 58y + 49
Trinomial w/ two binomial factors: Are there two numbers
whose product is 9  49 or 441 and whose sum is -58?
YES, the product of -9 and -49 is 441 and their sum is -58.
 9 1

 ( y  1)
9
1
-58
-49
-9
441
 49
 (9 y  49)
9
(9y – 49)(y – 1)
Check this by using
FOIL.
Ex. 3 – Factor by grouping
4m 4 n  6m 3 n 2  16m 2 n  24mn2
 2mn(2m 3  3m 2 n  8m  12n)
Pull GCF of 2mn
 2mn[( 2m 3  3m 2 n)  (8m  12n)] Group two commons together
 2mn[m 2 (2m  3n)  4(2m  3n)]
 2mn[( m 2  4)( 2m  3n)]
Pull m2 out of first group and -4
out of 2nd group.
Recognize Difference of squares
 2mn[( m  2)( m  2)( 2m  3n)]
Factor remaining and simplify.
You can check this by multiplying the
factors.
Chalkboard examples #1
•
Factor 5ax2 – 45a
5ax  45a
2
5a( x  9)
2
5a( x  3)( x  3)
Chalkboard examples #2
•
Factor 32x3 – 50x
32 x  50 x
3
2 x( x  25)
2
2 x( x  5)( x  5)
Chalkboard examples #3
•
Factor x4 + x3 – 12x2
x  x  12 x
4
3
2
x ( x  x  12)
2
2
x ( x  4)( x  3)
2
GCF of x2
Factors of -12 that
subtract to give
you 1.
Chalkboard examples #4
•
Factor 12a3 – 16a2 – 16a
12a  16a  16a
3
2
4a(3a  4a  4)
2
4a(3a  2)( a  2)
GCF of x2
Factors of -12 that
subtract to give
you -4.
CB ex. 4: Factor
•
•
2
3a
– 4a – 4
Trinomial w/ two binomial factors: Are there two numbers
whose product is 3  -4 or -12 and whose sum is -4?
YES, the product of -6 and 2 is -12 and their sum is -4.
6 2

 ( a  2)
3
1
-4
+2
-6
-12
2
 (3a  2)
3
(a - 2)(3a + 2)
Check this by using
FOIL.