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CHAPTER 6
Factoring and Quadratic Equations
b. The trinomial 12w 4 1 52w 2 1 35 is quadratic in form. Use substitution and then
factor the remaining quadratic. Using u 5 w 2, we get the following:
12w 4 1 52w 2 1 35 5 12 w 2 2 1 52w 2 1 35
5 12u 2 1 52 u 1 35
5 2u 1 7 6u 1 5 5 2w 2 1 7 6w 2 1 5 Substitute in u and factor.
u ⴝ w2
Replace u with w 2.
Check the factorization.
2w
2
1 7 6w 2 1 5 5 12w 4 1 10w 2 1 42w 2 1 35
5 12w 4 1 52w 2 1 35
PRACTICE PROBLEM FOR EXAMPLE 7
Factor 3a 10 2 5a 5 2 28.
TEACHING TIP This tool kit is meant
as a summary of the techniques covered
in Sections 6.1–6.3. Be sure to point out
the first step in the first row of the tool kit,
namely, factor out the GCF first.
Summary of Factoring Techniques (Factoring Tool Kit)
Now we will summarize our list of how to factor a polynomial expression using the
factoring tool kit below.
Factoring Tool Kit
GCF First
Factor out the GCF first, if there is
one. Always look for the GCF first!
6x3 10x2 22x
2x 3x2 5x 11 Section 6.1
Factor by grouping
12 x2 1 8 x 2 21x 2 14
5 4x 2 7 3x 1 2 Section 6.1
Coefficient of x2 is 1
Find the factors of c that sum to b
x2 1 7 x 1 12
5 x 1 3 x 1 4 Section 6.2
Coefficient of x2 is not 1
AC method
12 x2 1 8 x 2 15
5 2x 1 3 6x 2 5 Section 6.2
Perfect square trinomial
Use the patterns
x2 1 6 x 1 9 5 x 1 3 2
x2 2 10 x 1 25 5 x 2 5 2
Section 6.3
4x 4 2 11x 2 2 3
Section 6.3
Four Terms
Three Terms
a 2ab b a b
a2 2ab b2 a b 2
2
Quadratic form
2
2
AC method
au bu c
where u is an expression
2
5
4x 2
1
1 x2
2
3
Two Terms
Difference of squares
x2 2 25 5 x 1 5 x 2 5 Section 6.3
Section 6.3
a b 3 a b a ab b 2 x3 8
x 2 x 2 2x 4 Difference of cubes pattern
x 3 27
Section 6.3
Difference of squares pattern
a b 2
Sum of cubes
2
b a
b
Sum of cubes pattern
3
Difference of cubes
a
2
a b a b a ab b
3
64016_ch06_sec 6.3.indd 484
3
2
2
x 3 x 3x 9 2
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