Special Products of Polynomials

Another special binomial product is the product of a sum and a difference of terms. For example, let’s multiply the following binomials.

\(\begin{align*}(x+4)(x-4) & = x^2-4x+4x-16\\ & = x^2-16\end{align*}\)

Notice that the middle terms are opposites of each other, so they cancel out when we collect like terms. This always happens when we multiply a sum and difference of the same terms.

\(\begin{align*}(a+b)(a-b)&=a^2-ab+ab-b^2\\ & =a^2-b^2\end{align*}\)

When multiplying a sum and difference of the same two terms, the middle terms cancel out. We get the square of the first term minus the square of the second term. You should remember this formula.

Sum and Difference Formula: \(\begin{align*}(a+b)(a-b) = a^2-b^2\end{align*}\)

Example 2: Multiply the following binomial and simplify.

\(\begin{align*}(5x+9)(5x-9)\end{align*}\)

Solution: Use the above formula, substituting \(\begin{align*}a=5x\end{align*}\) and \(\begin{align*}b=9\end{align*}\). Multiply.

\(\begin{align*}(5x+9)(5x-9)=(5x)^2-(9)^2=25x^2-81\end{align*}\)