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
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

$\begin{aligned} \text { dividend } &=(\text { divisor } \cdot \text { quotient })+\text { remainder } \\ 178 &=(3 \cdot 59)+1 \\ &=177+1 \\ &=178 \end{aligned}$

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 $2 x^{3}-3 x^{2}+4 x+5$ by $x+2$ using the long division algorithm, it would look like this:

We have found

$\frac{2 x^{3}-3 x^{2}+4 x+5}{x+2}=2 x^{2}-7 x+18-\frac{31}{x+2}$

or

$2 x^{3}-3 x^{2}+4 x+5=(x+2)\left(2 x^{2}-7 x+18\right)-31$

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 $f(x)$.

HOW TO

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

EXAMPLE 1

Using Long Division to Divide a Second-Degree Polynomial

Divide $5 x^{2}+3 x-2$ by $x+1$.

Solution

The quotient is $5x−2$. The remainder is 0. We write the result as

$\frac{5 x^{2}+3 x-2}{x+1}=5 x-2$

or

$5 x^{2}+3 x-2=(x+1)(5 x-2)$

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 2

Using Long Division to Divide a Third-Degree Polynomial

Divide $6 x^{3}+11 x^{2}-31 x+15 \text { by } 3 x-2$

Solution

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

$\frac{6 x^{3}+11 x^{2}-31 x+15}{3 x-2}=2 x^{2}+5 x-7+\frac{1}{3 x-2}$

Analysis

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

$(3 x-2)\left(2 x^{2}+5 x-7\right)+1=6 x^{3}+11 x^{2}-31 x+15$

Notice, as we write our result,

TRY IT #1

Divide $16 x^{3}-12 x^{2}+20 x-3$ by $4 x+5$.