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Wize High School Grade 11 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 Polynomials in Quadratic Form

A community pool is being built, and here is the blueprint for the pool.

The pool is the shaded blue area. The inner and outer perimeter of this pool is a square.

a) Write an expression representing the area of the pool. Do NOT simplify anything!

b) Factor the expression representing the area of the pool. Simplify each factor as much as possible.

c) Expand and simplify the expression representing the area of the pool.

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