EDGE Math 8

Factoring Special Products

Perfect Square Trinomials
A trinomial is called a perfect square if it is in either of the following forms: $x^{2} + 2 x y + y^{2}$ or $x^{2} - 2 x y + y^{2} .$ Notice that the first term and the last term of a perfect square trinomial are perfect squares. Its middle term is twice the product of the square roots of the first term and the last term.

Perfect square trinomials can be factored as squares of binomials, as shown below.

x^{2} + 2 x y + y^{2} = {(x + y)}^{2}

x^{2} - 2 x y + y^{2} = {(x - y)}^{2}

Example

Factor each polynomial completely.

t^{2} - 2 t + 1

a^{2} + 10 a b + 25 b^{2}

4 c^{2} - 12 c d + 9 d^{2}

25 x^{2} + 30 x y + 9 y^{2}

a^{6} + 14 a^{3} b + 49 b^{2}

{(c + d)}^{2} - 16 (c + d) + 64

Solution

t^{2} - 2 t + 1 = {(t - 1)}^{2}

a^{2} + 10 a b + 25 b^{2}

= {(a + 5 b)}^{2}

4 c^{2} - 12 c d + 9 d^{2}

= {(2 c - 3 d)}^{2}

25 x^{2} + 30 x y + 9 y^{2}

= {(5 x + 3 y)}^{2}

a^{6} + 14 a^{3} b + 49 b^{2}

= {(a^{3} + 7 b)}^{2}

{(c + d)}^{2} - 16 (c + d) + 64

= {(c + d - 8)}^{2}