Use Long Division to Divide Polynomials

In this section, you will:

• Use long division to divide polynomials.
• Use synthetic division to divide polynomials.

Figure 1 Lincoln Memorial, Washington, D.C.

The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length $61.5$ meters (m), width $40$ m, and height $30$ m. We can easily find the volume using elementary geometry.

$\begin{aligned} V &=l \cdot w \cdot h \\ &=61.5 \cdot 40 \cdot 30 \\ &=73,800 \end{aligned}$

So the volume is $73,800$ cubic meters $\left(\mathrm{m}^{3}\right)$. Suppose we knew the volume, length, and width. We could divide to find the height.

$\begin{aligned} h &=\frac{V}{l \cdot w} \\ &=\frac{73,800}{61.5 \cdot 40} \\ &=30 \end{aligned}$

As we can confirm from the dimensions above, the height is $30$ m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial $3 x^{4}-3 x^{3}-33 x^{2}+54 x$. The length of the solid is given by $3x$; the width is given by $x−2$. To find the height of the solid, we can use polynomial division, which is the focus of this section.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-4-dividing-polynomials
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