6.3.3 Dividing Polynomials Using Synthetic Division

Part 1: Use Long Division

You are going to solve a division problem using long division.

Part 2: Use Synthetic Division

You are now going to solve the same division problem using synthetic division.

2. $(x^2-13x+42)÷(x-6)$

Compare your answer:

$(x^2-13x+42)÷(x-6)$

Step 1 - Write the dividend with decreasing powers of $x$. Make sure there is no degree missing from the highest degree down to a degree of $0$.

$x^2-13x+42$

Step 2 - Write the coefficients of the terms as the first row of the synthetic division.

Step 3 - Write the divisor as $x-c$ and place $c$ in the synthetic division in the divisor box.

Step 4 - Bring down the first coefficient to the third row.

Step 5 - Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.

Step 6 - Add the second column, putting the result in the third row.

Step 7 - Multiply that result by the divisor and place the result in the second row under the third coefficient.

Step 8 - Add the final column, putting the result in the third row.

Step 9 - The division is complete. The numbers in the third row give us the result. The $1$ $-7$ in the third row are the coefficients of the quotient. The quotient is $x-7$. The $0$ in the box in the third row is the remainder.

Dividing Polynomials Using Synthetic Division

Mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in a previous example and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

Synthetic division basically just removes unnecessary repeated variables and numbers. Here, all the $x$ and $x^2$ are removed as well as the $-x^2$ and $-4x$ because they are opposite the term above.

The first row of the synthetic division is the coefficients of the dividend. The $-5$ is the opposite of the $5$ in the divisor.

The second row of the synthetic division is the numbers shown in red in the division problem.

The third row of the synthetic division is the numbers shown in blue in the division problem.

Notice the quotient and remainder are shown in the third row.

Synthetic division only works when the divisor is of the form $x-c$.

The following example will explain the process.

Example 1

Use synthetic division to find the quotient and remainder when $2x^3+3x^2+x+8$ is divided by $x+2$.

Step 1 - Write the dividend with decreasing powers of $x$.

$2x^3+3x^2+x+8$

Step 2 - Write the coefficients of the terms as the first row of the synthetic division.

Step 3 - Write the divisor as $x-c$ and place $c$ in the synthetic division in the divisor box.

Step 4 - Bring down the first coefficient to the third row.

Step 5 - Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.

Step 6 - Add the second column, putting the result in the third row.

Step 7 - Multiply that result by the divisor and place the result in the second row under the third coefficient.

Step 8 - Add the third column, putting the result in the third row.

Step 9 - Multiply that result by the divisor and place the result in the third row under the third coefficient.

Step 10 - Add the final column, putting the result in the third row.

Step 11 - The quotient is $2x^2-1x+3$, and the remainder is $2$.

The division is complete. The numbers in the third row give us the result. The $2$ $-1$ $3$ are the coefficients of the quotient. The quotient is $2x^2-1x+3$. The $2$ in the box in the third row is the remainder.

Compare your answer:

$\begin{array}{}\\ \\ \hfill \text{(quotient)(divisor)}+\text{remainder}& =\hfill & \text{dividend}\hfill \\ \hfill \left(2x^2-1x+3\right)\left(x+2\right)+2& \stackrel{?}{=}\hfill & 2x^3+3x^2+x+8\hfill \\ \hfill 2x^3-x^2+3x+4x^2-2x+6+2& \stackrel{?}{=}\hfill & 2x^3+3x^2+x+8\hfill \\ \hfill 2x^3+3x^2+x+8& =\hfill & 2x^3+3x^2+x+8\✓\hfill \end{array}$

Example 2

Use synthetic division to find the quotient and remainder when $x^4-16x^2+3x+12$ is divided by $x+4$.

The polynomial $x^4-16x^2+3x+12$ has its terms in order with descending degree, but we notice there is no $x^3$ term. We will add a $0$ as a placeholder for the $x^3$ term. In $x-c$ form, the divisor is $x-(-4)$.

We divided a $4^{\text{th}}$ degree polynomial by a 1st degree polynomial, so the quotient will be a $3^{\text{rd}}$ degree polynomial. Reading from the third row, the quotient has the coefficients $1$ $-4$ $0$ $3$, which is $x^3-4x^2+3$. The remainder is $0$.