Transcript No Slide Title

Factoring - Perfect Square Trinomial
• A Perfect Square Trinomial is any trinomial that is the
result of squaring a binomial.
 x  3
2
Binomial
Squared
 x  6x  9
2
Perfect Square
Trinomial
• 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
a  2ab  b
 a  b
2
2
a  2ab  b
2
2
Note the pattern for the signs:
 a  b
2
2
• Here is how to identify a perfect square trinomial:
a  2ab  b
2
2
a  2ab  b
2
2
1. Both first and last terms are perfect squares
Note that there is always a positive sign on
both of these terms.
2. The middle term is given by
2ab
If these two conditions are met, then the
expression is a perfect square trinomial.
• Example 1
x  8x  16
2
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2ab ?
2
2
2
x  8x  16   x   8 x   4 
2. Is the middle term
2ab  2( x)(4)  8x
• Since the trinomial is a perfect square, factor it using
the pattern:
2
2
2
a  2ab  b   a  b 
 x
2
 8x   4
2
1. First term a:
(x
2. Last term b:
( x 4)
3. Sign same as
the middle term
( x  4)
4. Squared
( x  4)
2
• Example 2
x  10 x  25
2
Factor:
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Is the middle term
2ab ?
x  10 x  25   x   10 x   5 
2ab  2  x  5  10x
2
2
2
• Since the trinomial is a perfect square, factor it using
the pattern:
2
2
2
a  2ab  b   a  b 
 x
2
 10 x   5 
2
1. First term:
(x
2. Last term
( x 5)
3. Sign same as
the middle term
( x  5)
4. Squared
( x  5)
2
• Example 3
Factor:
4 x  12 x  9
2
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares?
2. Check the middle term:
2ab 
2 2x  3  12x
• Since the trinomial is a perfect square, factor it using
the pattern:
2
2
2
a  2ab  b   a  b 
4 x  12 x  9
2
1. First term:
(2x
2. Last term
(2 x 3)
3. Sign same as
the middle term
(2 x  3)
4. Squared
(2 x  3)
2
• Example 4
Factor:
4x  7 x  9
2
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  12x
This is not a perfect square trinomial. If it can be
factored, another method will have to be used.
No
• Example 5
Factor:
9 x  20 x  12
2
Determine if the trinomial is a perfect square
trinomial.
1. Are both first and last terms perfect squares? No
This is not a perfect square trinomial. If it can be
factored, another method will have to be used.
• Example 6
Factor:
x  10 x  25
2
Determine if the trinomial is a perfect square
trinomial.
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.
• Example 7
Factor:
25x  60 xy  36 y
2
2
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 6 y  60xy
• Since the trinomial is a perfect square, factor it using
the pattern:
2
2
2
a  2ab  b   a  b 
25x  60 xy  36 y
2
2
1. First term:
(5x
2. Last term
(5 x 6 y )
3. Sign same as
the middle term
(5x  6 y)
4. Squared
(5x  6 y)
2