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.

Difference of Squares Example Problems

Let's go through a couple of example problems together to practice using the difference of squares pattern.

Example 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).
3. The factored form of the polynomial x² – 49 is (x + 7)(x - 7).

Example 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.
5. The x-intercepts are located at x = 2 and x = -2.