Factoring Perfect Square
Trinomials
Factoring Perfect Square
Trinomials
What is a perfect square?
If a number is squared the result is a perfect
square.
Example 22=4
4 is a perfect square.
Other examples:
32=9 or 42=16
9 is a perfect square.
16 is a perfect square.
Factoring Perfect Square
Trinomials
Here is a list of the perfect squares for the numbers 130.
12=1
112=121
212=441
22=4
122=144
222=484
32=9
132=169
232=529
42=16
142=196
242=576
52=25
152=225
252=625
62=36
162=256
262=676
72=49
172=289
272=729
82=64
182=324
282=784
92=81
192=361
292=841
102=100
202=400
302=900
Factoring Perfect Square
Trinomials
When a variable is raised to an even power it is
a perfect square.
Example: (x)(x)= x2
x2 is a perfect square.
(x3)(x3)= x6 or (x5)(x5)= x10
x6 and x10 are both perfect squares.
Factoring Perfect Square
Trinomials
If a number or a variable is a perfect square the square
root of the quantity is the number or variable that
was squared to get the perfect square.
Example: Square 9.
9x9 = 81
81 is the perfect square.
9 is the square root of 81.
Example: Square x3
(x3) (x3) = x6 or (x3)2 = x6
x6 is the perfect square.
x3 is the square root of x6
Factoring Perfect Square
Trinomials
• Now we are ready to understand the termperfect square trinomial.
• The trinomial that results from squaring a
binomial is a perfect square trinomial.
• Example: (x+7)2 = x2+14x+49
• x2+14x+49 is a perfect square trinomial.
• We know that a perfect square trinomial
always results when a binomial is squared.
• The reverse is also true.
• When we factor a perfect square trinomial
the result is always a squared binomial.
Factoring Perfect Square
Trinomials
• Here are few examples:
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Factor: x2+10x+25
Result: (x+5)2
Check by multiplying
x2+10x+25
Factor: x2+2xy+y2
(x+y) 2
Check by multiplying.
x2+2xy+y2
Factoring Perfect Square
Trinomials
• Not all trinomials are perfect square trinomials.
• How do we recognize that a trinomial is a perfect square
trinomial.
• The first and last terms of the trinomial must be perfect squares
and must be positive.
• Example: x2+10x+25
• What about the middle term? +10x
• Take the square root of the first term x2 and get x.
• Take the square root of the last term +25 and get 5.
• Multiply (5)(x) and double the result. 10x. That is your middle
term.
• Two times the product of the square roots of the first and last
terms will give the middle term.
Factoring Perfect Square
Trinomials
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Here are some examples of trinomials that are perfect square trinomials.
4x2 -20x +25
2x
5
(2x- 5)2
9x2 - 48xy + 64y2
3x
8y
(3x-8y)2
2x3 +20x2y+50xy2
Factor out the GCF
2x(x2+10xy+25y2)
2x
x
5y
2x(x+5y)2
Factoring Perfect Square
Trinomials
Here are some examples that are not perfect square trinomials.
x2+10x-25
The last term is not positive.
x2+2xy+2y2
The 2 in the last term is not a perfect square.
4x2-10xy+25y2
The square root of the first term is 2x.
The square root of the last term 5y.
2(2x)(5y)= 20xy
20xy = 10xy
4x2-16xy+8y2
There is a common factor of four.
4(x2- 4xy + 2y2)
The last term of the trinomial is not a perfect square because the 2 in the last term
is not perfect square.
To get more help go to the tutorial Practice- Factoring Perfect Square Trinomials