Algebra

##### Related Lessons

Tutors/teachers: Nikkolas
Tutor and Aerospace Engineer

# Difference of Squares

## The Difference of Squares Pattern

There are special patterns that some polynomials follow. The difference of squares pattern is when a polynomial is made up of a square minus another square. This pattern allows us to easily factor the polynomial. The formula is given as:
a2 – b2 = (a + b)(a – b)

Where a and b may be any algebraic expression. For example, a could be a variable such as “y”, and b could be a number such as “5”. Or, a could be a more complicated expression such as “2x3“. The difference of squares pattern can be used for any set of algebraic expressions.

### When to use the Difference of Squares

The difference of squares pattern comes in handy when we must factor a polynomial to find the roots/zeros or to simplify it. For example, we might be tasked with finding the roots/zeros of the equation y = x2 – 9.

We can use the difference of squares to factor the right-hand side of the equation and then find the roots/zeroes. Applying the difference of squares pattern to x2 – 9 gives us (x + 3)(x – 3). Using this factored form, we can determine that the roots/zeros are x = -3 and x = 3.

### Example Problems

Problem 1:
Using the difference of squares pattern, factor the polynomial x2 – 49.

Solution:
1.) Let’s set a = x and b = 7 because a2 = x2 and b2 = 49.
2.) Applying the pattern (a + b)(a – b), we get (x + 7)(x – 7).
The factored form of the polynomial x2 – 49 is (x + 7)(x – 7).

Problem 2:
Find the x-intercepts of y = 4x4 – 64.

Solution:
1.) We find the x-intercepts by setting y = 0. This gives us 0 = 4x4 – 64.
2.) We must now find the zeroes of the expression 4x4 – 64. The expression is a difference of squares binomial.
3.) Applying the formula, we get:
a2 – b2 = (a + b)(a – b)
4x4 – 64 = (2x2 + 8)(2x2 – 8)
4.) Now that it’s in factored form, we can find the zeroes. (2x2 + 8)(2x2 – 8) has zeroes at x = 2 and x = -2.
The x-intercepts are located at x = 2 and x = -2.

Result :

Worksheet 1