Factoring: Special Cases
Warm Up
• Factor the following:
1. x2 + 2x – 15
2. 5x2 - 17x + 6
3. x2 + 6x + 9
Today’s Objectives
• Students will continue to factor polynomials
and look at special cases
Case 1: “Difference of Squares”
• a binomial with both terms
being squares:
•
a  b  (a  b)( a  b)
2
2
Factoring a Difference of Squares
• Let’s use our box method to find the
factors of a difference of squares
• Example: x2 – 16
– Why is this a “difference of squares”?
To factor a difference of squares,
express each term as a square of a
monomial then apply the rule...
2
2
a  b  (a  b)(a  b)
2
Ex: x 16 
2
2
x 4 
(x  4)(x  4)
Here is another
example:
x - 81=
2
x -9 =
2
2
( x + 9) ( x - 9)
But what if we had:
1 2
x  81 
49
2
1 x  92  1 x  91 x  9
7
7

7 
Try these on your own:
1. x  121
2
  x  11 x  11
2. 9y  169x
2
3. x  16
4
2
  3 y  13 x  3 y  13 x 
 ( x  4)( x  4)
2
 ( x  4)( x  2)( x  2)
2
Be careful!
2
Case 2: “Sum and Difference of
Cubes”
a  b  a  ba  ab  b
3
3
a b
3
3

 a  ba  ab  b 
2
2
2
2
Write each term as a cube and apply either
of the rules.
Example :
Rewrite as cubes
3
3
x  64  (x  4 )
3
Apply the rule for sum of cubes:
a  b  a  ba  ab  b
3
3
2
2
2
 (x  4)(x  x 4  4 )
2
 (x  4)(x  4x  16)
2

Rewrite as cubes
3
3
3
Ex: 8y  125  ((2y)  5 )
Apply the rule for difference of cubes:
a  b  a  ba  ab  b
3
3
2

2

 2y  5 2y  2y  5  5
2
2
 2y  54y  10y  25
2

Try these on your own:
2
1. x  5x  6
2
2. 3x  11x  20
3
3. x  216
4. 8x  8
3
5. 3x  6x  24x
3
2
Answers:
1. (x  6)(x  1)
2. (3x  4)(x  5)
2
3. (x  6)(x  6x  36)
4. 8(x  1)(x  x  1)
2
5. 3x(x  4)(x  2)
Scavenger Hunt
Homework