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*}
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