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

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:

• 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.

• Example 12 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 28 4x x

2ab 2( )(4)x 8x

• 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. Squared 2( 4)x

22 22a ab b a b

2 28 4x x

• Example 22 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 2 210 5x x

2ab 2 5x 10x

• 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. Squared 2( 5)x

22 22a ab b a b

2 210 5x x

• 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

• 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. Squared 2(2 3)x

22 22a ab b a b 24 12 9x x

• 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.

• 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

• 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.

• 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

• 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. Squared 2(5 6 )x y

22 22a ab b a b 2 225 60 36x xy y