What you’ll learn to do: Use multiple techniques to divide polynomials

The Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)

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.^[1] We can easily find the volume using elementary geometry.

[latex]\begin{array}{l}V=l\cdot w\cdot h\hfill \\ \text{}V=61.5\cdot 40\cdot 30\hfill \\ \text{}V=73,800\hfill \end{array}[/latex]

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

[latex]\begin{array}{l}h=\frac{V}{l\cdot w}\hfill \\ \text{}h=\frac{73,800}{61.5\cdot 40}\hfill \\ \text{}h=30\hfill \end{array}[/latex]

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 [latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x[/latex]. The length of the solid is given by 3x; the width is given by [latex]x - 2[/latex]. To find the height of the solid, we can use polynomial division, which is the focus of this section.