Download Factoring Special Cases

Lesson 9.6
Perfect Squares and
Factoring, pg. 508
Objectives:
To factor perfect square trinomials.
To solve equations involving perfect
squares.
Difference of Squares
2
a
2
b
- = (a - b)(a + b)
or
2
2
a - b = (a + b)(a - b)
The order does not matter!!
4 Steps for factoring
Difference of Squares
1. Are there only 2 terms?
2. Is the first term a perfect square?
3. Is the last term a perfect square?
4. Is there subtraction (difference) in the
problem?
If all of these are true, you can factor
using this method!!!
1. Factor x2 - 25
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
 Two
x2 – 25
terms?Yes
Yes
 1st term a perfect square?
Yes
 2nd term a perfect square?
 Subtraction?Yes
( x + 5 )(x - 5
 Write your answer!
2. Factor 16x2 - 9
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
 Two
16x2 – 9
terms?Yes
Yes
 1st term a perfect square?
Yes
 2nd term a perfect square?
 Subtraction?Yes
(4x + 3 )(4x- 3 )
 Write your answer!
3. Factor 81a2 – 49b2
When factoring, use your factoring table.
Do you have a GCF? No
Are the Difference of Squares steps true?
 Two
81a2 – 49b2
terms?Yes
Yes
 1st term a perfect square?
Yes
 2nd term a perfect square?
 Subtraction?Yes
(9a + 7b )(9a- 7b
)
 Write your answer!
Factor x2 – y2
1.
2.
3.
4.
(x + y)(x + y)
(x – y)(x + y)
(x + y)(x – y)
(x – y)(x – y)
Remember, the order doesn’t matter!
REVIEW
The Square of a Binomial

Multiply (m – 4)2.

Multiply (5n + 3)2.
What to look for…
Ax2 + Bx + C
 Clue 1: A & C are positive, perfect
squares.
 Clue 2: B is the square root of A times
the square root of C, doubled.
If these two things are true, the trinomial is
a Perfect Square Trinomial and can be
factored as (x + y)2 or (x – y)2.
General Form of Perfect
Square Trinomials
 x2
+ 2xy + y2 = (x + y)2
or
 x2 – 2xy + y2 = (x - y)2
 Note:
When factoring, the sign
in the binomial is the same as
the sign of B in the trinomial.
Just watch and think.





Ex) x2 + 12x + 36
What’s the square root
of A? of C?
Multiply these and
double. Does it = B?
Then it’s a Perfect
Square Trinomial!
Solution: (x + 6)2

Ex) 16a2 – 56a + 49
Square root of A?
of C?
Multiply and
double…
= B?

Solution: (4a – 7) 2



Ex. 1: Determine whether each
trinomial is a perfect square
trinomial. If so, factor it.
1. y² + 8y + 16
2. 9y² - 30y + 10
Example 2: Factoring perfect
square trinomials.

1) x2 + 8x + 16

3) 4z2 – 36z + 81
2) 9n2 + 48n + 64
4) 9g² +12g - 4
4)
25x² - 30x + 9
6) 49y² + 42y + 36
5)
x² + 6x - 9
7) 9m³ + 66m² - 48m
General Rules for Factoring
Factor out the GCF, if possible.
2)
Is the polynomial a BINOMIAL?
a) Is it the Difference of Two Perfect Squares?
x² - y² = (x – y)(x + y)
3)
Is the polynomial a TRINOMIAL?
a) Is it a perfect square trinomial?
a² + 2ab + b² = (a + b) ²
a² - 2ab + b² = (a – b) ²
b) Is it a trinomial with leading coefficient 1?
-Find two numbers with a sum of B and product of C
c) Is it a trinomial with leading coefficient other than 1?
-Find two numbers with a sum of B and product of AC
-Replace the middle term with the two numbers and
factor by grouping.
4) Four or more terms, FACTOR by GROUPING
1)
Ex. 3: Factor completely.
1) 2x² + 18
3) 5a³ - 80a
2) c² - 5c + 6
4) 8x² - 18x - 35
Ex. 3: Solve each equation.
1) 3x² + 24x + 48 = 0
2) 49a² + 16 = 56a
3) z² + 2x + 1= 16
4) (y – 8)² = 7