9.7
Factoring Special Cases
9.7 – Factoring Special Cases
Goals / “I can…”
Factor perfect – square trinomials
Factor the difference of squares
9.7 – Factoring Special Cases
First of all, what do these numbers
have in common?
4
9
16
81
64
25
49
144
121
225
9.7 – Factoring Special Cases
Recall:
2
2
(x + 2) = x + 4x + 4
2
2
(x – 2) = x – 4x + 4
These are called perfect square
trinomials.
Factoring: Perfect Square Trinomials
The first criteria of a
Perfect Square Trinomial
is that it must have three
terms.
Using FOIL we find the product of
two binomials.
(a + b)(a + b)
= a + ab + ab + b
2
2
= a + 2ab + b
2
2
Rewrite the perfect square trinomial
as a binomial squared.
So when you recognize this…
a + 2ab + b = (a + b)(a + b)
2
= ( a + b)
2
2
…you can write this.
Recognizing a Perfect Square Trinomial
x + 10 x + 25
2
First term must be a perfect square.
(x)(x) = x2
Last term must be a perfect square.
(5)(5) = 25
Middle term must be twice the product
of the square root coefficient of the first
and last term.
(2)(5)(1) = 10
What if it is a Perfect Square Trinomial
x + 10 x + 25
2
If you have a perfect Square Trinomial it is
easy to factor:
Take the square root of the first term.
Take the square root of the last term.
Use the sign of the middle term, put in
parenthesis and square the result.
2
2
x + 10 x + 25 = ( x + 5)
Recognizing a Perfect Square Trinomial
m + 8m + 16 = (m + 4)
2
2
First term must be a perfect square.
(m)(m) = m2
Last term must be a perfect square.
(4)(4) = 16
Middle term must be twice the product
of the square root coefficient of the first
and last term.
(2)(4)(1) = 8
Recognizing a Perfect Square Trinomial
p − 18 p + 81 = ( p − 9)
2
2
First term must be a perfect square.
(p)(p) = p2
Last term must be a perfect square.
(9)(9) = 81
Signs must match!
Middle term must be twice the
product of the coefficient of the first
and last term.
(2)(9)(1) = 18p
Recognizing a Perfect Square Trinomial
121 p + 110 p + 100 Not a Perfect
2
Square
First term must be a perfect square.
(11p)(11p) = 121p2Trinomial!
Last term must be a perfect square.
(10)(10) = 100
Middle term must be twice the product of the
first and last term.
(2)(10)(11p) = 220p
Perfect Square Trinomial Y/N
Yes (r − 4 )
r − 8r + 16
2
2
Yes (7 p − 2 )
49 p − 28 p + 4
2
49 s − 42 st + 36t
2
2
4m + 4mn + n
2
d + 50d + 225
2
2
2
No
Yes (2m + n )
2
No
9.7 – Factoring Special Cases
2
(x + 2)(x – 2) = x – 4
This is called the difference of squares
What characteristics to both have in
common? Different?
Factoring Special Cases:
Difference of Squares
Difference of Squares
a2 – b2 = (a + b)(a - b)
Factoring Special Cases:
Difference of Squares
A binomial that is created by subtracting two perfect
squares.
Ex1: x2 – 4
Ex2: x2 – 625
Ex3: 4x2 – 25
Ex4: 16x2 - 81
What is true about the factored form of each of these
binomials?
Factoring Special Cases:
Difference of Squares
Factor each of the following completely:
1) x2 – 100
2) x4 – 16
3) 100x2 – 400
4) 3x2 - 75
5) 225x2 – 121y2
#2 isn’t quadratic and #5 isn’t a polynomial, but
you can still factor them!
9.7 – Factoring Special Cases
Using these, we can look to factor
polynomials.
2
4t + 36t + 81
- Notice that the first
and last terms are
perfect squares. It
has 3 terms, so it is a
perfect square trinomial.
9.7 – Factoring Special Cases
2
9v – 16
- Notice that the first
and last terms are
perfect squares, but it
has 2 terms, so it
is the difference of
squares.
9.7 – Factoring Special Cases
REMEMBER, sometimes we need to
factor a GCF first
3m 3 – 12m
I can’t factor an m3,
but I can factor a 3m
2
3m(m – 4)
3m(m – 2)(m + 2)
Now I can factor.