CP Algebra I
Homework: ________________________________________
8.7 Factoring Special Cases
Objectives: To factor perfect square trinomials and difference of two squares
Any trinomial of the form 𝑎2 + 2𝑎𝑏 + 𝑏 2 or 𝑎2 − 2𝑎𝑏 + 𝑏 2 is a perfect square trinomial
because it is the result of squaring a binomial.
Example) Multiply: (𝑥 + 4)2 = 𝑥 2 + 4𝑥 + 4𝑥 + 16 = 𝑥 2 + 8𝑥 + 16
Here is how to recognize a perfect square trinomial:
 The first and last term are perfect squares
 The middle term is twice the product of one factor from the first term and one factor
from the last term
Factoring a Perfect Square Trinomial
Example 1) To factor you reverse the multiplication process: 𝑥 2 + 10𝑥 + 25
Take the square root of the first term: _________________
Take the square root of the last term: _________________
Multiply these two together & double it to check the middle term: _______________________
The factored form is _________________________________
Note: the sign of the middle term in the given determines the sign of the binomial
If the 10𝑥 is NOT the right middle term, then the polynomial can’t be factored into the
square of a binomial.
Example 2) Factor 4𝑥 2 − 48𝑥 + 36
Take the square root of the first term: _________________
Take the square root of the last term: _________________
Multiply these two together & double it to check the middle term: _______________________
The factored form is _________________________________
Quick Check: Factor each perfect square trinomial
1) 9𝑥 2 + 18𝑥 + 9
2) 𝑥 2 − 14𝑥 + 49
Factor the trinomial square into the square of a binomial:
3) 𝑛2 + 18𝑛 + 81
4) 𝑔2 − 6𝑔 + 9
5) 𝑚2 + 12𝑚 + 36
6) 16𝑥 2 + 8𝑥 + 1
7) 1 − 2𝑣 + 𝑣 2
8) 𝑟 2 + 25 − 10𝑟
Factoring to Find a Length
Example 3) Digital images are composed of thousands of tiny pixels rendered as squares, as
shown below. Suppose the area of a pixel is 4𝑥 2 + 20𝑥 + 25. What is the length of one side of
the pixel?
Quick Check: You are building a square patio. The area of the patio is 16𝑚2 − 72𝑚 + 81. What
is the length of one side of the patio?
Factoring a Difference of Two Squares
Consider the binomials: (𝑥 + 3)(𝑥 − 3) =
𝑥 2 − 3𝑥 + 3𝑥 − 9 =
𝑥2 − 9
Note: When multiplying, the middle term will always cancel out because they are opposite
each other. The resulting product is called the difference of two squares. Note that each term
is a perfect square.
The “sum of two squares” is prime and cannot be factored: 𝑥 2 + 25 = CANNOT be factored
Factoring a Difference of Two Squares: For all real numbers 𝑎 and 𝑏, 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
Example 4) What is the factored form of 16𝑥 2 − 81?
Take the square root of the first term: _______________________
Take the square root of the second term: ____________________
Use the rule for the difference of squares to write your final answer: ________________________
Example 5) What is the factored form of 25𝑥 2 − 64?
Take the square root of the first term: _______________________
Take the square root of the second term: ____________________
Use the rule for the difference of squares to write your final answer: ________________________
Example 6) The expression 25𝑥 2 + 64 contains two perfect squares. Can you use the same
method as above to factor it? Explain your reasoning.
Quick Check: Factor each of the following using a difference of squares.
9) 𝑥 2 − 16
10) 𝑟 2 − 9
12) 81𝑥 2 − 100
13) 16𝑦 2 − 49
14) 4𝑥 2 − 25
15) 36𝑥 2 − 4
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Factoring Out A Common Factor: When you factor out the GCF of a polynomial sometimes the
expression that remains is a perfect square trinomial or the difference of two squares. You can then
factor this expression further using the rules from this lesson.
Example 7) Factor 24𝑔2 − 6
Example 8) Factor 12𝑥 2 + 12𝑥 + 3
Quick Check: Factor each expression completely
16) 12𝑥 2 − 48
17) 8𝑥 2 + 32𝑥 + 32