# Difference of Squares

## Related Calculator: Factoring Polynomials Calculator

Difference of squares (something squared minus something else squared):

 huge color(purple)(a^2-b^2=(a-b)(a+b))

Proof of this fact is straightforward.

We just prove it from right to left.

Apply FOIL to (a-b)(a+b):

(a-b)(a+b)=a*a+a*b+(-b)*a+(-b)*b=a^2+ab-ab-b^2=a^2-b^2.

This formula can also be derived by applying techniques from Factoring Quadratics note.

Note: formula is valid only for a^2-b^2. Sum of squares a^2+b^2 can't be factored at all.

Example 1. Factor x^2-9.

Notice, that 9=3^2.

Thus, x^2-9=x^2-3^2=(x-3)(x+3).

Answer: x^2-9=(x-3)(x+3).

Of course, there can be more complex expressions.

Example 2. Factor 9y^2-49.

Notice, that 9y^2=(3y)^2 and 49=7^2.

Thus, 9y^2-49=(3y)^2-7^2=(3y-7)(3y+7).

Answer: 9y^2-49=(3y-7)(3y+7).

And even harder...

Example 3. Factor the following: (x+y)^2-25u^4b^6.

Notice, that 25u^4b^6=(5u^2b^3)^2.

Thus, (x+y)^2-25u^4b^6=(x+y)^2-(5u^2b^3)^2=((x+y)-5u^2b^3)((x+y)+5u^2b^3).

Answer: (x+y)^2-25u^4b^6=(x+y-5u^2b^3)(x+y+5u^2b^3).

Now, it is time to exercise.

Exercise 1. Factor the following: n^2-36.

Answer: (n-6)(n+6).

Exercise 2. Factor the following: -1+49x^2.

Answer: (7x-1)(7x+1). Hint: -1+49x^2=49x^2-1.

Exercise 3. Factor 144c^10d^8-(m+n)^2.

Answer: (12c^5d^4-m-n)(12c^5d^4+m+n).

Exercise 4. Factor (x+y)^2-(x-y)^2.

Answer: 4xy.