Carson Elementary And Intermediate Algebra 3e
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Carson Elementary and Intermediate Algebra 3e
Section 6.4
Objectives
1.
Factor perfect square trinomials.
2.
Factor a difference of squares.
3.
Factor a difference and sum of cubes.
Vocabulary
conjugates
Binomials that differ only in the sign separating the terms.
Prior Knowledge
2
2
2
(a + b)
= (a + b)(a + b) = a
+ 2ab + b
2
2
(a + b)(a – b) = a
– b
Multiplying polynomials
The first 20 perfect squares and 5 perfect cubes.
New Concepts
This section shows several special cases for factoring. This is really a memorizing activity. You must
recognize the special case and then know that it will factor a certain way.
These are the special cases:
2
2
2
Perfect Square Trinomial:
x
+ 2xy + y
= (x + y)
2
2
2
– 2xy + y
= (x – y)
x
When you see that the first and last terms are perfect squares, there’s a good chance the trinomial is a
perfect square trinomial that will factor by this rule. Always foil back to be sure it works.
2
2
Example 1:
y
+ 6y + 9
(y + 3)
2
2
2
– 20ab + 25b
(2a – 5b)
Example 2:
4a
2
– 24m + 16
2
(3m – 4)
Example 3:
9m
2
Example 4:
y
+ 6y + 36
Not a perfect square
2
2
2
– 12by + 9b
(2y – 3b)
Example 5:
4y
2
2
– 72q + 24
6(3q – 2)
Example 6:
54q
2
2
– b
= (a – b)(a + b)
Difference of two squares:
a
What you need to recognize in this special case is that two perfect squares are subtracted. There is no
middle term. The only way the middle term of the trinomial will disappear is if, when you foil, the inner
and outer terms are the same but with different signs. We call these conjugates.
2
2
The sum of two squares is prime and cannot be factored: a
+ b
is prime.
2
– 121
(a + 11)(a – 11)
Example 7:
a
2
– 16
(5x – 4)(5x + 4)
Example 8:
25x
2
49 – a
(7 – a)(7 + a)
Example 9:
2
2
– 36y
4(4m – 3y)(4m + 3y)
Example 10:
64m
V. Zabrocki 2011
page 1
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