Multiplying a pair of binomials can become a tedious and time-consuming process. Fortunately, there is a clever technique that allows us to easily identify patterns and efficiently expand these products. These patterns are referred to as special products and can significantly simplify our calculations.

A special product is the result of multiplying two binomials that follow a predictable pattern.

Before delving into the topic of special products, let's first review the FOIL method and its application in multiplying binomials.

The FOIL method is a helpful tool for multiplying binomials. In case you need a refresher, here is the formula:

(a + b) * (c + d) = ac + ad + bc + bd

Let's practice with a few examples. Can you identify any patterns when expanding these expressions?

- (x + 3) * (x - 3)
- (2a + 5) * (2a - 5)
- (3b + 2) * (3b + 2)
- (4x - 7) * (4x + 7)

Did you notice that the middle term cancels out in the first two examples and is twice the product of the terms in the last two examples? These are all special products, each following a specific rule that simplifies our calculations. These shortcuts come in handy when working with polynomials, making it easier to expand and factorize them. Our learning objectives for this topic are as follows:

- Identifying patterns in special products

- Using these patterns to expand and factorize polynomials

- Solving polynomials using these factoring techniques

In the following section, we will explore each type of pattern in detail. The three main types are:

- Difference of two squares
- Perfect square trinomials
- Sum and difference of two cubes

We will then apply these methods to solve polynomials, just as we did in the previous section.

The general formula for the sum and difference of two terms is:

(a + b)(a - b) = a^{2}- b^{2}

While this may seem confusing, let's visualize it geometrically. Look at the square below:

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