table of contents factoring - perfect square trinomial a perfect square trinomial is any trinomial...
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Table of Contents
Factoring - Perfect Square Trinomial
• A Perfect Square Trinomial is any trinomial that is the result of squaring a binomial.
23x
Binomial Squared
2 6 9x x
Perfect Square Trinomial
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2 22a ab b
• Our goal now is to start with a perfect square trinomial and factor it into a binomial squared. Here are the patterns.
Perfect Square Trinomial
Factored
2a b
2 22a ab b 2a b
Note the pattern for the signs:
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• Here is how to identify a perfect square trinomial:
1. Both first and last terms are perfect squares
2 22a ab b 2 22a ab b
2. The middle term is given by 2ab
If these two conditions are met, then the expression is a perfect square trinomial.
Note that there is always a positive sign on both of these terms.
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• Example 1
2 8 16x x Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
2. Is the middle term 2 ?ab2 8 16x x 2 2
8 4x x
2ab 2( )(4)x 8x
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• Since the trinomial is a perfect square, factor it using the pattern:
1. First term a:
2. Last term b:
(x
( 4)x
3. Sign same as the middle term
( 4)x
4. Squared2( 4)x
22 22a ab b a b
2 28 4x x
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• Example 2
2 10 25x x Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
2. Is the middle term 2 ?ab2 10 25x x 2 2
10 5x x 2ab 2 5x 10x
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• Since the trinomial is a perfect square, factor it using the pattern:
1. First term:
2. Last term
(x
( 5)x
3. Sign same as the middle term
( 5)x
4. Squared2( 5)x
22 22a ab b a b
2 210 5x x
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• Example 3
Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term:
2 2x 3
24 12 9x x
12x
2ab
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• Since the trinomial is a perfect square, factor it using the pattern:
1. First term:
2. Last term
(2x
(2 3)x
3. Sign same as the middle term
(2 3)x
4. Squared2(2 3)x
22 22a ab b a b 24 12 9x x
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• Example 4
Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term: 2 2x 3
24 7 9x x
12x No
This is not a perfect square trinomial. If it can be factored, another method will have to be used.
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• Example 5
Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
29 20 12x x
This is not a perfect square trinomial. If it can be factored, another method will have to be used.
No
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• Example 6
Factor:
Determine if the trinomial is a perfect square trinomial.
2 10 25x x
This is not a perfect square trinomial since the last term has a negative sign.
Perfect square trinomials always have a positive sign for the last term.
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• Example 7
Factor:
Determine if the trinomial is a perfect square trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term:
2 5x 6y
2 225 60 36x xy y
60xy
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• Since the trinomial is a perfect square, factor it using the pattern:
1. First term:
2. Last term
(5x
(5 6 )x y
3. Sign same as the middle term
(5 6 )x y
4. Squared2(5 6 )x y
22 22a ab b a b 2 225 60 36x xy y
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