5.5: Dividing Polynomials

Learning Objectives

• Use long division to divide polynomials.

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.

\[\begin{align*} V&=l \; {\cdot} \; w \; {\cdot} \; h \\ &=61.5 \; {\cdot} \; 40 \; {\cdot} \; 30 \\ &=73,800 \end{align*}\]

So the volume is \(73,800\) cubic meters (\(m^3\)).

Figure \(\PageIndex{1}\): Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)

Suppose we knew the volume, length, and width. We could divide to find the height.

\[\begin{align*} h&=\dfrac{V}{l{\cdot}w} \\&=\dfrac{73,800}{61.5{\cdot}40} \\ &=30 \end{align*}\]

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 \(3x^4−3x^3−33x^2+54x\). 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.

Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.

Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.

\[\begin{align*} \text{dividend}&=(\text{divisor}{\cdot}\text{quotient})+\text{remainder} \\ 178&=(3{\cdot}59)+1 \\ &=177+1 \\ &=178\end{align*}\]

We call this the Division Algorithm and will discuss it more formally after looking at an example.

Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide \(2x^3−3x^2+4x+5\) by \(x+2\) using the long division algorithm, it would look like this:

\[\require{enclose}\begin{array} {rll} \large x+2\enclose{longdiv}{ 2x^3-3x^2+4x+5\phantom{0}} & \qquad &\large\text{Set up the division problem.}\\[8pt]
\large 2x^2 \hspace{5.45em} & \qquad & \large2x^3\text{ divided by }x\text{ is }2x^2.\\[-3pt]
\large x+2\enclose{longdiv}{ 2x^3-3x^2+4x+5\phantom{0}} & \qquad &\\[8pt]
\large 2x^2 \hspace{5.45em} & \qquad & \\[-3pt]
\large x+2\enclose{longdiv}{ 2x^3-3x^2+4x+5\phantom{0}} & \qquad &\\[-3pt]
\large \underline{-\left( 2x^3 + 4x^2 \right)} \hspace{4.9em} & \qquad & \large\text{Multiply }x+2\text{ by }2x^2\text{ and subtract.}\\[-3pt]
\large -7x^2+4x \hspace{2.8em} & \qquad & \large\text{Bring down the next term.} \\[8pt]

\large 2x^2 - 7x\hspace{2.85em} & \qquad & \large -7x^2\text{ divided by }x\text{ is }-7x.\\[-3pt]
\large x+2\enclose{longdiv}{ 2x^3-3x^2+4x+5\phantom{0}} & \qquad &\\[-3pt]
\large \underline{-\left( 2x^3 + 4x^2 \right)} \hspace{4.9em} & \qquad & \\[-3pt]
\large -7x^2+4x \hspace{2.8em} & \qquad & \\[-3pt]
\large \underline{-\left( -7x^2 - 14x \right)} \hspace{2.0em} & \qquad & \large\text{Multiply }x+2\text{ by }-7x.\\[-3pt]
\large 18x+5 \phantom{0} & \qquad & \large\text{Subtract and bring down the next term.} \\[8pt]

\large 2x^2 - 7x+18 & \qquad & \large 18x\text{ divided by }x\text{ is }18.\\[-3pt]
\large x+2\enclose{longdiv}{ 2x^3-3x^2+4x+5\phantom{0}} & \qquad &\\[-3pt]
\large \underline{-\left( 2x^3 + 4x^2 \right)} \hspace{4.9em} & \qquad & \\[-3pt]
\large -7x^2+4x \hspace{2.8em} & \qquad & \\[-3pt]
\large \underline{-\left( -7x^2 - 14x \right)} \hspace{2.0em} & \qquad & \\[-3pt]
\large 18x+\phantom{0}5 & \qquad & \\[-3pt]
\large \underline{-\left( 18x + 36 \right)} \hspace{-0.45em} & \qquad & \large\text{Multiply }x+2\text{ by }18.\\[-3pt]
\large -31 & \qquad & \large\text{Subtract.} \\[8pt]
\end{array} \nonumber\]

We have found

\[\dfrac{2x^3−3x^2+4x+5}{x+2}=2x^2−7x+18−\dfrac{31}{x+2} \nonumber\]

or

\[ 2x^3−3x^2+4x+5=(x+2)(2x^2−7x+18)−31 \nonumber\]

We can identify the dividend, the divisor, the quotient, and the remainder.

Writing the result in this manner illustrates the Division Algorithm.

The Division Algorithm

The Division Algorithm states that, given a polynomial dividend \(f(x)\) and a non-zero polynomial divisor \(d(x)\) where the degree of \(d(x)\) is less than or equal to the degree of \(f(x)\), there exist unique polynomials \(q(x)\) and \(r(x)\) such that

\[f(x)=d(x)q(x)+r(x)\]

\(q(x)\) is the quotient and \(r(x)\) is the remainder. The remainder is either equal to zero or has degree strictly less than \(d(x)\).

If \(r(x)=0\), then \(d(x)\) divides evenly into \(f(x)\). This means that, in this case, both \(d(x)\) and \(q(x)\) are factors of \(f(x)\).

Given a polynomial and a binomial, use long division to divide the polynomial by the binomial

1. Set up the division problem.
2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
3. Multiply the answer by the divisor and write it below the like terms of the dividend.
4. Subtract the bottom binomial from the top binomial.
5. Bring down the next term of the dividend.
6. Repeat steps 2–5 until reaching the last term of the dividend.
7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.

Example \(\PageIndex{1}\): Using Long Division to Divide a Second-Degree Polynomial

Divide \(5x^2+3x−2\) by \(x+1\).

Solution

The quotient is \(5x−2\). The remainder is 0. We write the result as

\[\dfrac{5x^2+3x−2}{x+1}=5x−2 \nonumber\]

or

\[5x^2+3x−2=(x+1)(5x−2) \nonumber\]

Analysis

This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.

Example \(\PageIndex{2}\): Using Long Division to Divide a Third-Degree Polynomial

Divide \(6x^3+11x^2−31x+15\) by \(3x−2\).

Solution

\[\require{enclose}\begin{array} {rll}
\large 2x^2 + \phantom{0}5x-\phantom{0}7 & \qquad & \large 6x^3\text{ divided by }3x\text{ is }2x^2.\\[-3pt]
\large 3x-2\enclose{longdiv}{ 6x^3+11x^2-31x+15} & \qquad &\\[-3pt]
\large \underline{-\left( 6x^3 - 4x^2 \right)} \hspace{5.8em} & \qquad & \large\text{Multiply }3x-2\text{ by }2x^2. \\[-3pt]
\large 15x^2-31x \hspace{3.0em} & \qquad & \large\text{Subtract. Bring down next term. }15x^2\text{ divided by }3x\text{ is }5x. \\[-3pt]
\large \underline{-\left( 15x^2 - 10x \right)} \hspace{2.5em} & \qquad & \large\text{Multiply }3x-2\text{ by }5x. \\[-3pt]
\large -21x+15 \hspace{0.5em}& \qquad & \large\text{Subtract. Bring down next term. }-21x\text{ divided by }3x\text{ is }-7. \\[-3pt]
\large \underline{-\left( -21x + 14 \right)} \hspace{0.1em} & \qquad & \large\text{Multiply }3x-2\text{ by }-7.\\[-3pt]
\large 1 \hspace{0.5em}& \qquad & \large\text{Subtract. The remainder is 1.} \\[8pt]
\end{array} \nonumber\]

There is a remainder of 1. We can express the result as:

\[\dfrac{6x^3+11x^2−31x+15}{3x−2}=2x^2+5x−7+\dfrac{1}{3x−2} \nonumber\]

Analysis

We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.

\[(3x−2)(2x^2+5x−7)+1=6x^3+11x^2−31x+15 \nonumber\]

Notice, as we write our result,

• the dividend is \(6x^3+11x^2−31x+15\)
• the divisor is \(3x−2\)
• the quotient is \(2x^2+5x−7\)
• the remainder is \(1\)

Try It! \(\PageIndex{2}\)

Divide \(16x^3−12x^2+20x−3\) by \(4x+5\).

Solution

\(4x^2−8x+15−\dfrac{78}{4x+5}\)

Using Polynomial Division to Solve Application Problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.

Example \(\PageIndex{6}\): Using Polynomial Division in an Application Problem

The volume of a rectangular solid is given by the polynomial \(3x^4−3x^3−33x^2+54x\). The length of the solid is given by \(3x\) and the width is given by \(x−2\). Find the height \(h\) of the solid.

Solution

There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in Figure \(\PageIndex{3}\). Let \(h\) equal the height of the box.

Figure \(\PageIndex{3}\)

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

\[\begin{align*} V&=l{\cdot}w{\cdot}h \\ 3x^4−3x^3−33x^2+54x&=3x{\cdot}(x−2){\cdot}h \end{align*}\]

To solve for \(h\), first divide both sides by \(3x\).

\[\dfrac{3x{\cdot}(x−2){\cdot}h}{3x}=\dfrac{3x^4−3x^3−33x^2+54x}{3x} \nonumber\]

\[(x-2)h=\dfrac{x^3-x^2-11x+18}{x-2} \nonumber\]

Now solve for \(h\) using synthetic division.

\[h=\dfrac{x^3−x^2−11x+18}{x−2} \nonumber\]

\[ \large{\begin{array}{c}-2\\ \\ \\
\end{array}}
{\begin{align*}&\\[0pt]
&{\begin{array}{r|}\\[0pt]
\\[0pt] \end{array}}\\[1pt]
& \\[2pt]& \end{align*}}\!\!
{\begin{array}{rrrr}
1 & -1 & -11 & 18 \\
& 2 & 2 & -18 \\
\hline 1 & 1 & -9 & 0
\end{array}}
\nonumber\]

The quotient is \(x^2+x−9\) and the remainder is \(0.\) The height of the solid is \(x^2+x−9\).

Try It! \(\PageIndex{4}\)

The area of a rectangle is given by \(3x^3+14x^2−23x+6\). The width of the rectangle is given by \(x+6\). Find an expression for the length of the rectangle.

Solution

\(3x^2−4x+1\)

Key Equations

Division Algorithm \(f(x)=d(x)q(x)+r(x)\) where \(q(x){\neq}0\)

Key Concepts

• Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.
• The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
• Polynomial division can be used to solve application problems, including area and volume.

Footnotes

\(^1\) National Park Service. "Lincoln Memorial Building Statistics." www.nps.gov/linc/historycultu...statistics.htm. Accessed 4/3/2014

Glossary

Division Algorithm

given a polynomial dividend \(f(x)\) and a non-zero polynomial divisor \(d(x)\) where the degree of \(d(x)\) is less than or equal to the degree of \(f(x)\), there exist unique polynomials \(q(x)\) and \(r(x)\) such that \(f(x)=d(x)q(x)+r(x)\) where \(q(x)\) is the quotient and \(r(x)\) is the remainder. The remainder is either equal to zero or has degree strictly less than \(d(x)\).