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# Difference of Squares

## Difference of Squares Lesson

### 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.

INTRODUCING

#### 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.
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