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Wize High School Grade 10 Math Textbook > Factoring Polynomials

Factoring Difference of Squares

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Factoring a Difference of Squares

There is a special type of quadratic (degree 2) polynomial called a difference of squares, it has the form a2−b2\large a^2-b^2, and it turns out there is a short-cut to factoring these polynomials.

Example
Factor a2−b2a^2-b^2.

We can rewrite this as a2+0a−b2a^2+0a-b^2.


So, the short-cut for factoring a difference of squares is

a2−b2=(a−b)(a+b)\Large\boxed{\colorFive{a}^2-\colorFour{b}^2=(\colorFive{a}-\colorFour{b})(\colorFive{a}+\colorFour{b})}

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Example: Factoring a Difference of Squares

Factor the following polynomials fully.

a) x2−9x^2-9

Notice that we have a difference of squares

x2−32=(x−3)(x+3)\begin{aligned} &x^2-3^2\\ =&(x-3)(x+3) \end{aligned}



b) 16−x216-x^2

Notice that we have a difference of squares

42−x2=(4−x)(4+x)\begin{aligned} &4^2-x^2\\ =&(4-x)(4+x) \end{aligned}




c) 4y2−254y^2-25

Notice that we have a difference of squares

(2y)2−52=(2y−5)(2y+5)\begin{aligned} &(2y)^2-5^2\\ =&(2y-5)(2y+5) \end{aligned}

Practice: Factoring a Difference of Squares

Factor these polynomials.

a) x2−100x^2-100

b) t2−36t^2-36

Practice: Factoring a Difference of Squares

Factor the following polynomials.

a) 9m2−499m^2-49

b) 4c2−81d24c^2-81d^2

Practice: Factoring a Difference of Squares

Factor the following polynomials fully.

a) 3x2−123x^2-12

b) 45cx2−20cy245cx^2-20cy^2

c) 64x4−9y264x^4-9y^2

Practice: Factoring a Difference of Squares

Factor the following polynomial fully.

a) x2−25x^2-25

b) t2−81t^2-81

c) 18m2−3218m^2-32

d) 500c2−5500c^2-5

e) 4x2−y24x^2-y^2

f) 3m4−27n43m^4-27n^4