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Wize High School Grade 10 Math Textbook > Factoring Polynomials

Factoring Perfect Square Trinomials

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Factoring Perfect Square Trinomials

There is a special type of quadratic (degree 2) polynomial called a perfect square trinomial, it has the following forms
a2+2ab+b2\large a^2+2ab+b^2 or a2−2ab+b2\large a^2-2ab+b^2
It turns out there is a short-cut to factoring these polynomials.

Example
Factor a2+2ab+b2a^2+2ab+b^2 and a2−2ab+b2a^2-2ab+b^2.




So, the short-cut for factoring a perfect square trinomial is:
a2  +  2ab  +  b2   =   (a  +  b) 2\Large\boxed{\colorFive{a}^2 ~~ + ~~ 2\colorFive{a}\colorFour{b}~~+~~\colorFour{b}^2~~~=~~~(\colorFive{a}~~+~~\colorFour{b})~^2}
and
a2  −  2ab  +  b2   =   (a − b) 2\Large\boxed{\colorFive{a}^2~~\colorbox{yellow}{$-$}~~2\colorFive{a}\colorFour{b}~~+~~\colorFour{b}^2~~~=~~~(\colorFive{a}~\colorbox{yellow}{$-$}~\colorFour{b})~^2}

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Example: Factoring Perfect Square Trinomials

Factor the following polynomials fully.

a) x2+6x+9x^2+6x+9

Notice that we have a perfect square trinomial:
x2+2(x)(3)+32=(x+3)2\begin{aligned} &\bcfi{x}^2+2(\bcfi{x})(\bcf{3})+\bcf{3}^2\\[1em] =&(\bcfi{x}+\bcf{3})^2 \end{aligned}

So, the factored form is (x+3)2\boxed{(x+3)^2}.


b) x2−8x+16x^2-8x+16

Notice that we have a perfect square trinomial:
x2−2(x)(4)+42=(x−4)2\begin{aligned} &\bcfi{x}^2-2(\bcfi{x})(\bcf{4})+\bcf{4}^2\\[1em] =&(\bcfi{x}-\bcf{4})^2 \end{aligned}

So, the factored form is (x−4)2\boxed{(x-4)^2}.

Practice: Factoring Perfect Square Trinomials

Factor the following polynomials fully.

a) x2+2dx+d2x^2+2dx+d^2

b) x2−2cx+c2x^2-2cx+c^2

Practice: Factoring Perfect Square Trinomials

Factoring the following polynomials fully.

a) x2−10x+25x^2-10x+25

b) t2+4t+4t^2+4t+4

Practice: Factoring Perfect Square Trinomials

Factoring the following polynomials fully.

a) 3x2+30x+753x^2+30x+75

b) −2x2+12x−18-2x^2+12x-18

c) 4x2+12xy+9y24x^2+12xy+9y^2

Practice: Factoring Perfect Square Trinomials

Factor the following polynomials fully.

a) x2+12x+36x^2+12x+36

b) x2−16x+64x^2-16x+64

c) 4x2−4x+14x^2-4x+1

d) 3x2+18x+273x^2+18x+27

e) −32x2+80x−50-32x^2+80x-50