Use Synthetic Division to Divide Polynomials

As we've seen, 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 the section.

Divide $2 x^{3}-3 x^{2}+4 x+5$ by $x+2$ using the long division algorithm.

The final form of the process looked like this:

There is a lot of repetition in the table. 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.

Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. 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 $–2$, multiply and add. The process starts by bringing down the leading coefficient.

We then multiply it by the "divisor" and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is $2 x^{2}-7 x+18$ and the remainder is $–31$. The process will be made more clear in Example 3.

SYNTHETIC DIVISION

Synthetic division is a shortcut that can be used when the divisor is a binomial in the form $x−k$ where $k$ is a real number. In synthetic division, only the coefficients are used in the division process.

HOW TO

Given two polynomials, use synthetic division to divide.

EXAMPLE 3

Using Synthetic Division to Divide a Second-Degree Polynomial

Use synthetic division to divide $5 x^{2}-3 x-36 \text { by } x-3$.

Solution

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

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

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.

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.

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

EXAMPLE 4

Using Synthetic Division to Divide a Third-Degree Polynomial

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

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

Analysis

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

Figure 2

EXAMPLE 5

Using Synthetic Division to Divide a Fourth-Degree Polynomial

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

Solution

Notice there is no $x$-term. We will use a zero as the coefficient for that term.

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

TRY IT #2

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

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.

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 6

Using Polynomial Division in an Application Problem

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 $3x$ 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 3.

Figure 3

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

$\begin{aligned} V &=l \cdot w \cdot h \\ 3 x^{4}-3 x^{3}-33 x^{2}+54 x &=3 x \cdot(x-2) \cdot h \end{aligned}$

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

$\begin{aligned} &\frac{3 x \cdot(x-2) \cdot h}{3 x}=\frac{3 x^{4}-3 x^{3}-33 x^{2}+54 x}{3 x} \\ &(x-2) h=x^{3}-x^{2}-11 x+18 \end{aligned}$

Now solve for $h$ using synthetic division.

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

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

TRY IT #3

The area of a rectangle is given by $3 x^{3}+14 x^{2}-23 x+6$. The width of the rectangle is given by $x+6$. Find an expression for the length of the rectangle.