Example 6.5.18
Simplify the expression \((x+5)^3\) into an expanded polynomial.
Before we start expanding this expression, it is important to recognize that \((x+5)^3\neq x^3+ 5^3\text{.}\) We can see that this doesn't work by inputting \(1\) for \(x\) and applying the order of operations:
\begin{align*}
(\substitute{1}+5)^3\amp=6^3\amp \substitute{1}^3+5^3\amp=1+125\\
\amp=216\amp\amp=126
\end{align*}
With this in mind, we will need to rely on distribution to expand this expression. The first step in expanding \((x+5)^3\) is to remember that the exponent of \(3\) indicates that
\begin{equation*}
(x+5)^3=\overbrace{(x+5)(x+5)(x+5)}^{3\text{ times }}
\end{equation*}
Once we rewrite this in an expanded form, we next multiply the two binomials on the left and then finish by multiplying that result by the remaining binomial:
\begin{align*}
(x+5)^3\amp=\highlight{\left[(x+5)(x+5)\right]}(x+5)\\
\amp=\highlight{\left[x^2+10x+25\right]}(x+5)\\
\amp=x^3 + 5x^2 + 10x^2 +50x+25x+125\\
\amp=x^3 + 15x^2 + 75x + 125
\end{align*}