Q. 285 - Intermediate Algebra

Question

Multiply the following:

$$\begin{gathered} \left(x+2\right)\left(x-2\right) \\ \left(y+7\right)\left(y-7\right) \\ \left(w+5\right)\left(w-5\right) \end{gathered}$$

Explain the pattern that you see in your answers.

Step-by-Step Solution

Answer

The product is $$\begin{gathered} x^2-4 \\ {y}^2-49 \\ {w}^2-25 \end{gathered}$$

The pattern seen in the answers is the Product of Conjugates pattern.

Step 1. Given Information

$$\begin{gathered} \left(x+2\right)\left(x-2\right) \\ \left(y+7\right)\left(y-7\right) \\ \left(w+5\right)\left(w-5\right) \end{gathered}$$

Step 2. Multiply the conjugates

$$\begin{gathered} \left(x+2\right)\left(x-2\right) \\ {x}^2-4 \\ \left(y+7\right)\left(y-7\right) \\ {y}^2-49 \\ \left(w+5\right)\left(w-5\right) \\ {w}^2-25 \end{gathered}$$

These are conjugates. They have the same first numbers and the same last numbers, and one binomial is a sum and the other is a difference.

It fits the Product of Conjugates pattern.