Dividing Polynomials

By the end of this section, you will be able to:

Use long division to divide polynomials
Use synthetic 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. We can easily find the volume using elementary geometry.

\begin{array}{lrcl} & V &=& l\cdot w\cdot h \\[4pt] & &=& 61.5\cdot40\cdot30 \\[4pt] & &=& 73{,}800 \end{array}

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

\begin{array}{lrcl} & h &=& \tfrac{V}{l\cdot w} \\[4pt] & &=& \tfrac{73{,}800}{61.5\cdot40} \\[4pt] & &=& 30 \end{array}

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, and we can 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.

\begin{array}{r} \phantom{3\,\overline{\smash{)}\,}}\phantom{1}59 \\[-1pt] 3\,\overline{\smash{)}\,178} \\[-1pt] \phantom{3\,\overline{\smash{)}\,}}\underline{-15}\phantom{8} \\[-1pt] \phantom{3\,\overline{\smash{)}\,}}\phantom{-}28 \\[-1pt] \phantom{3\,\overline{\smash{)}\,}}\underline{-27} \\[-1pt] \phantom{3\,\overline{\smash{)}\,}}\phantom{-2}1 \end{array}

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{array}{lrcl} & \text{dividend} &=& (\text{divisor}\cdot\text{quotient})+\text{remainder} \\[4pt] & 178 &=& (3\cdot59)+1 \\[4pt] & &=& 177+1 \\[4pt] & &=& 178 \end{array}

We call this the Division Algorithm, and we 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:

\begin{array}{r} 2x^2-7x+18\phantom{{}+5}\phantom{)} \\[3pt] x+2\,\overline{\smash{)}\,2x^3-3x^2+4x+5}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}2x^3+4x^2\mathrlap{)}}\phantom{{}+4x+5}\phantom{)} \\[3pt] -7x^2+4x\phantom{{}+5}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}{-7x^2-14x}\mathrlap{)}}\phantom{{}+5}\phantom{)} \\[3pt] 18x+5\phantom{)} \\[3pt] \underline{\mathllap{-\,(}18x+36\mathrlap{)}}\phantom{)} \\[3pt] -31\phantom{)} \end{array}

We have found

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

2x^3-3x^2+4x+5=(x+2)(2x^2-7x+18)-31.

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

the dividend is $2x^3-3x^2+4x+5$
the divisor is $x+2$
the quotient is $2x^2-7x+18$
the remainder is $-31$

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)

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 $d(x)$ . If $r(x)=0$ , then $d(x)$ divides evenly into $f(x)$ , and 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.

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

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

Solution.

\begin{array}{r} 5x-2\phantom{{}-2}\phantom{)} \\[2pt] x+1\,\overline{\smash{)}\,5x^2+3x-2}\phantom{)} \\[2pt] \underline{-\left(5x^2+5x\right)}\phantom{{}-2} \\[2pt] -2x-2\phantom{)} \\[2pt] \underline{-\left(-2x-2\right)} \\[2pt] 0\phantom{)} \end{array}

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

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

5x^2+3x-2=(x+1)(5x-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. Divide $6x^3+11x^2-31x+15$ by $3x-2$ .

Solution.

\begin{array}{r} 2x^2+5x-7\phantom{{}+15}\phantom{)} \\[3pt] 3x-2\,\overline{\smash{)}\,6x^3+11x^2-31x+15}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}6x^3-4x^2\mathrlap{)}}\phantom{{}-31x+15}\phantom{)} \\[3pt] 15x^2-31x\phantom{{}+15}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}15x^2-10x\mathrlap{)}}\phantom{{}+15}\phantom{)} \\[3pt] -21x+15\phantom{)} \\[3pt] \underline{\mathllap{-\,(}{-21x+14}\mathrlap{)}}\phantom{)} \\[3pt] 1\phantom{)} \end{array}

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

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

Analysis. We can check our work by using the Division Algorithm to rewrite the solution, then multiplying.

(3x-2)(2x^2+5x-7)+1=6x^3+11x^2-31x+15

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 $1$

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

Using synthetic division to divide polynomials

Long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is $1$ .

To illustrate the process, recall the example at the beginning of this subsection: dividing $2x^3-3x^2+4x+5$ by $x+2$ using the long division algorithm. The final form of the process looked like this:

\begin{array}{r} 2x^2-7x+18\phantom{{}+5}\phantom{)} \\[3pt] x+2\,\overline{\smash{)}\,2x^3-3x^2+4x+5}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}2x^3+4x^2\mathrlap{)}}\phantom{{}+4x+5}\phantom{)} \\[3pt] -7x^2+4x\phantom{{}+5}\phantom{)} \\[3pt] \underline{\mathllap{-\,(}{-7x^2-14x}\mathrlap{)}}\phantom{{}+5}\phantom{)} \\[3pt] 18x+5\phantom{)} \\[3pt] \underline{\mathllap{-\,(}18x+36\mathrlap{)}}\phantom{)} \\[3pt] -31\phantom{)} \end{array}

There is a lot of repetition in that tableau. If we don’t write the variables but instead line up their coefficients in columns under the division sign, and also eliminate the partial products, we already have a simpler version of the entire problem.

\begin{array}{r} 2\,\overline{\smash{)}\,2\quad{-3}\quad\ \,4\quad\ \,5\,} \\[3pt] \phantom{2\,\overline{\smash{)}\,}}\underline{{-2}\quad{-4}}\phantom{\quad\ \,4\quad\ \,5\,} \\[3pt] \phantom{2\,\overline{\smash{)}\,2\quad{-3}\quad\ \,}}\underline{{-7}\quad14}\phantom{\quad\ \,5\,} \\[3pt] \phantom{2\,\overline{\smash{)}\,2\quad{-3}\quad\ \,4\quad\ \,}}\underline{18\quad{-36}} \\[3pt] \phantom{2\,\overline{\smash{)}\,2\quad{-3}\quad\ \,4\quad\ \,5\,}}{-31} \end{array}

Synthetic division carries this simplification a few steps further: collapse the table by moving each of the rows up to fill any vacant spot. Also, instead of dividing by $2$ — as we would in division of whole numbers — then multiplying and subtracting the middle product, we change the sign of the “divisor” to $-2$ and multiply and add. The process starts by bringing down the leading coefficient, then multiplying it by the “divisor” and adding, repeating this process column by column until there are no entries left. The bottom row holds the coefficients of the quotient, and its last entry is the remainder:

\begin{array}{r|rrrr} -2 & 2 & -3 & 4 & 5 \\ & & -4 & 14 & -36 \\ \hline & 2 & -7 & 18 & -31 \end{array}

The quotient is $2x^2-7x+18$ and the remainder is $-31$ — the same result the long-division tableau gave above.

Synthetic division. Synthetic division is a shortcut that can be used when the divisor is a binomial in the form

x-k

. In synthetic division, only the coefficients are used in the division process.

How to: given two polynomials, use synthetic division to divide.

Write $k$ for the divisor.
Write the coefficients of the dividend.
Bring the lead coefficient down.
Multiply the lead coefficient by $k$ . Write the product in the next column.
Add the terms of the second column.
Multiply the result by $k$ . Write the product in the next column.
Repeat steps 5 and 6 for the remaining columns.
Use the bottom numbers to write the quotient. The number in the last column is the remainder. The next number from the right has degree $0$ , the next number has degree $1$ , and so on.

Example. Use synthetic division to divide $5x^2-3x-36$ by $x-3$ .

Solution. Begin by setting up the synthetic division. Write $k=3$ and the coefficients.

\begin{array}{r|rrr} 3 & 5 & -3 & -36 \end{array}

Bring down the lead coefficient. Multiply the lead coefficient by $k$ .

\begin{array}{r|rrr} 3 & 5 & -3 & -36 \\ & & 15 & \\ \hline & 5 & & \end{array}

Continue by adding the numbers in the second column. Multiply the resulting number by $k$ . Write the result in the next column. Then add the numbers in the third column.

\begin{array}{r|rrr} 3 & 5 & -3 & -36 \\ & & 15 & 36 \\ \hline & 5 & 12 & 0 \end{array}

The result is $5x+12$ . The remainder is $0$ . So $x-3$ is a factor of the original polynomial.

Analysis. Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.

(x-3)(5x+12)+0=5x^2-3x-36

Example. Use synthetic division to divide $4x^3+10x^2-6x-20$ by $x+2$ .

Solution. The binomial divisor is $x+2$ , so $k=-2$ . Add each column, multiply the result by $-2$ , and repeat until the last column is reached.

\begin{array}{r|rrrr} -2 & 4 & 10 & -6 & -20 \\ & & -8 & -4 & 20 \\ \hline & 4 & 2 & -10 & 0 \end{array}

The result is $4x^2+2x-10$ . The remainder is $0$ . Thus, $x+2$ is a factor of $4x^3+10x^2-6x-20$ .

Analysis. The graph of the polynomial function $f(x)=4x^3+10x^2-6x-20$ below shows a zero at $x=k=-2$ . This confirms that $x+2$ is a factor of $4x^3+10x^2-6x-20$ .

Figure: The graph of f of x equals 4x cubed plus 10x squared minus 6x minus 20, on a grid with x from negative 5 to 5 and y from negative 22 to 14. The curve crosses the x-axis at x equals negative 2, dips to a local minimum near x equals 1, then rises steeply to the right.

Example. Use synthetic division to divide $-9x^4+10x^3+7x^2-6$ by $x-1$ .

Solution. Notice there is no $x$ -term, so we will use a zero as the coefficient for that term.

\begin{array}{r|rrrrr} 1 & -9 & 10 & 7 & 0 & -6 \\ & & -9 & 1 & 8 & 8 \\ \hline & -9 & 1 & 8 & 8 & 2 \end{array}

The result is $-9x^3+x^2+8x+8+\tfrac{2}{x-1}$ .

Use synthetic division to divide $3x^4+18x^3-3x+40$ by $x+7$ .

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.

Example. 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 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.

Figure: An oblique sketch of a rectangular box representing the rectangular solid. Its bottom front edge is labeled Length 3x, its bottom edge receding into the page is labeled Width x - 2, and its right vertical edge is labeled Height.

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

\begin{array}{lrcl} & V &=& l\cdot w\cdot h \\[4pt] & 3x^4-3x^3-33x^2+54x &=& 3x\cdot(x-2)\cdot h \end{array}

To solve for $h$ , first divide both sides by $3x$ .

\begin{array}{lrcl} & \tfrac{3x\cdot(x-2)\cdot h}{3x} &=& \tfrac{3x^4-3x^3-33x^2+54x}{3x} \\[6pt] & (x-2)h &=& x^3-x^2-11x+18 \end{array}

Now solve for $h$ using synthetic division.

h=\frac{x^3-x^2-11x+18}{x-2}

\begin{array}{r|rrrr} 2 & 1 & -1 & -11 & 18 \\ & & 2 & 2 & -18 \\ \hline & 1 & 1 & -9 & 0 \end{array}

The quotient is $x^2+x-9$ and the remainder is $0$ . The height of the solid is $x^2+x-9$ .

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.

Key equations

Division Algorithm	$f(x)=d(x)q(x)+r(x)$ , where $q(x)\neq0$

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.
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form $x-k$ .
Polynomial division can be used to solve application problems, including area and volume.

Key terms

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 $f(x)=d(x)q(x)+r(x)$ ; $q(x)$ is the quotient and $r(x)$ is the remainder, and the remainder is either equal to zero or has degree strictly less than $d(x)$ . synthetic division — a shortcut method that can be used to divide a polynomial by a binomial of the form $x-k$ .

Practice

Use long division to divide polynomials

Use long division to divide $3x^2+23x+14$ by $x+7$ . What is the quotient?

Now give the remainder from dividing $3x^2+23x+14$ by $x+7$ .

Use long division to divide $2x^2-3x+2$ by $x+2$ . What is the quotient?

Now give the remainder from dividing $2x^2-3x+2$ by $x+2$ .

Use synthetic division to divide polynomials

Use synthetic division to find the quotient when $2x^3-6x^2-7x+6$ is divided by $x-4$ .

Use synthetic division to find the quotient when $x^3-21x^2+147x-343$ is divided by $x-7$ .

Is $x-2$ a factor of $3x^4-6x^3-5x+10$ ?

This section is adapted from Precalculus 2e, Section 3.5: Dividing Polynomials by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the decorative Lincoln Memorial photograph, which carries no mathematics; recreated every long-division and synthetic-division tableau as an accessible KaTeX array with \phantom spacing rather than as an image, staging each one exactly as the source’s image sequence does — the full long-division tableau, then the same work with the variables dropped, then the collapsed, sign-flipped synthetic-division grid bridging the two methods; and Example 3’s three partial grids (set up, bring down the lead coefficient, then complete the remaining columns), each paired with the source’s own narration; recreated the zero-confirming graph in Example 4’s Analysis as an accessible SVG plotted directly from $f(x)=4x^3+10x^2-6x-20$ , and omitted its inset close-up detail as decorative; recreated the labelled rectangular-solid sketch in the application example as an accessible SVG figure, an oblique box with its length, width, and height edges labelled to match the source; omitted the media links; converted the “Try It” problems into interactive fill-in exercises with instant feedback, each carrying an expanded answer form because “Divide $A$ by $B$ ” is otherwise satisfied by retyping the printed quotient $\tfrac{A}{B}$ ; and adapted five selected end-of-section exercises — two “specify the quotient and the remainder” long-division items, each split into paired quotient/remainder prompts, two “find the quotient” synthetic-division items, and a synthetic-division factor check — into seven interactive components in a closing Practice block, one group per objective.

Graphs of Polynomial Functions Zeros of Polynomial Functions