Dividing Polynomials — Long Division, Synthetic Division & More

In this tutorial on dividing polynomials, we’ll look at how to divide a polynomial by a monomial or by another polynomial. We’ll explore long division, synthetic division, and more.

But first, let’s make sure we have the basics covered.

The Basics

Here’s what a polynomial looks like. This is a polynomial with three terms - a trinomial.

3x^5y^2 - xy + 1

The parts of the polynomial separated by the addition/subtraction symbols are called terms. And each term in a polynomial can be seen as having two parts – the coefficient and the variables (along with their exponents).

Anatomy of an algebraic term — Anatomy of a term

Also, a polynomial having only a single term is known as a monomial. For example, $\hspace{0.2em} 2a^5 \hspace{0.2em}$ and $\hspace{0.2em} 9x^4y^5 \hspace{0.2em}$ .

Before you can divide polynomials you must be comfortable multiplying and dividing monomials. The good thing is — it’s really easy.

Multiplying/Dividing Monomials

To multiply monomials, multiply the numeric coefficients and add the exponents on identical (same) variables.

For example —

\begin{align*} {\color{Teal} x} ^ {\color{Red} 4} \cdot 2 {\color{Teal} x} ^ {\color{Red} 7} \, \, &= \, \, 2 {\color{Teal} x} ^{( {\color{Red} 4} + {\color{Red} 7} )} \\[1em] &= \, \, 2 {\color{Teal} x} ^ {\color{Red} 11} \end{align*}

Here’s another example.

And remember, if a variable doesn’t have a visible exponent, its exponent is $\hspace{0.2em} 1 \hspace{0.2em}$ .

\begin{align*} 3 {\color{Red} x} {\color{Teal} y^2} \cdot 5 {\color{Red} x^3} {\color{Teal} y} \, \, &= \, \, 3 \cdot 5 \cdot {\color{Red} x^1} \cdot {\color{Red} x} \cdot {\color{Teal} y^2} \cdot {\color{Teal} y} \\[1em] &= \, \, 15 \cdot {\color{Red} x^{(1 + 3)}} \cdot {\color{Teal} y^{(2 + 1)}} \\[1em] &= \, \, 15 {\color{Red} x^4} {\color{Teal} y^3} \end{align*}

To divide monomials, divide the numeric coefficients and take the difference of the exponents on identical (same) variables.

For example —

\begin{align*} \frac{12 {\color{Teal} x} ^{ {\color{Red} 5} }}{4 {\color{Teal} x} ^{ {\color{Red} 3} }} \, \, &= \, \, \frac{12}{4} \cdot {\color{Teal} x} ^{ {\color{Red} 5 - 3} } \\[1em] &= \, \, 3 {\color{Teal} x} ^ {\color{Red} 2} \end{align*}

Dividing Polynomials by Monomials

To divide a polynomial by a monomial, we distribute the division across the terms of the polynomial. And then, it's about dividing monomials by monomials.

The following examples should make it clear.

Examples

Simplify.

\begin{align*} &(i) \hspace{1.1em} 6x^4y - 12x^2y^2 - 3xy^3 \, \div \, 3x \\[1em] &(ii) \hspace{0.9em} a^3b^2c + 5b^2c^2 \, \div \, bc \end{align*}

Solution ( $\hspace{0.2em} i \hspace{0.2em}$ )

The question asks us to divide a trinomial by a monomial. We'll do it in two steps.

Step 1. The first step is to distribute the division across the three terms.

\begin{align*} &\frac{6x^4y - 12x^2y^2 - 3xy^3}{ {\color{Red} 3x} } \\[1.5em] = \hspace{0.5em}& \frac{6x^4y}{ {\color{Red} 3x} } - \frac{12x^2y^2}{ {\color{Red} 3x} } - \frac{3xy^3}{ {\color{Red} 3x} } \end{align*}

Step 2. Next, we do the division and simplication for each term. It would involve dividing monomials by monomials, something we've already seen.

\begin{align*} &\frac{6x^4y}{ {\color{Red} 3x} } - \frac{12x^2y^2}{ {\color{Red} 3x} } - \frac{3xy^3}{ {\color{Red} 3x} } \\[1.5em] = \hspace{0.5em} & \hspace{0.1em} 2x^3y \hspace{0.15em} - \hspace{0.55em} 4xy^2 \hspace{0.55em} - \hspace{0.7em} y^3 \end{align*}

That's it.

\hspace{0.2em} 2x^3y - 4xy^2 - y^3 \hspace{0.2em}

is our answer.

Solution ( $\hspace{0.2em} ii \hspace{0.2em}$ )

Nothing too different here. The same two steps.

\begin{align*} \frac{a^3b^2c + 5b^2c^2}{ {\color{Red} bc} } \hspace{0.5em} &= \hspace{0.5em} \frac{a^3b^2c}{ {\color{Red} bc} } + \frac{5b^2c^2}{ {\color{Red} bc} } \\[1.5em] &= \hspace{0.9em} a^2b \hspace{0.7em} + \hspace{0.55em} 5bc \end{align*}

Great. Now that we have the basics out of the way, time to dive into the real stuff.

Dividing Polynomials by Polynomials

When it comes to dividing polynomials by other polynomials, the most popular method is long division. So let’s start with that.

We have complete tutorials on long division and synthetic division methods with several examples to help you master each method. What follows is a quick overview of the two methods to help you get started.

Long Division

Long division for polynomials is quite similar to the long division method for numbers. Let’s see how it works with the following example.

Example

Find, using polynomial long division, the quotient and remainder when $\hspace{0.2em} 2x^3 + 8x^2 + 3x + 12 \hspace{0.2em}$ is divided by $\hspace{0.2em} x + 4 \hspace{0.2em}$ .

Solution

Step 1. Make sure both the divisor and the dividend are in their standard forms. Meaning, the terms in each polynomial should be arranged in decreasing order of their degrees.

Step 2. Write the division problem in the following manner.

Step 3. Divide the first term of the dividend by the first term of the divisor and write it in the top area.

Step 4. Multiply the result from step 3 with the whole divisor and write it below the dividend.

Step 5. Subtract the result from the dividend.

Step 6. If the degree of the difference is greater than or equal to the degree of the divisor, the difference becomes your new dividend. Repeat steps 3 to 6.

Here, the degree of the difference ( $\hspace{0.2em} 3x + 12 \hspace{0.2em}$ ) is $\hspace{0.2em} 1 \hspace{0.2em}$ . So, jump back to step 3.

Once the degree is less than that of the divisor, stop. If something non-zero remains at the bottom, it is the remainder.

Synthetic Division

Synthetic division is another method you can use for dividing polynomials. It’s not very popular but nonetheless, it is taught in schools and is interesting to know.

Note — We use synthetic division only for cases where the divisor is a linear polynomial of the form $\hspace{0.2em} x \pm a \hspace{0.2em}$ . For example, $\hspace{0.2em} x + 5 \hspace{0.2em}$ or $\hspace{0.2em} x - 2 \hspace{0.2em}$ .

Let me explain the method using the following examples.

Example

Find, using polynomial long division, the quotient and remainder when $\hspace{0.2em} 2x^3 + 8x^2 + 3x + 12 \hspace{0.2em}$ is divided by $\hspace{0.2em} x + 4 \hspace{0.2em}$ .

Solution

Step 1. Make sure the divisor is written as $\hspace{0.2em} x - k \hspace{0.2em}$ . Identify $\hspace{0.2em} k \hspace{0.2em}$ .

Step 2. Make sure the dividend polynomial is written in the standard form (write any missing terms with $\hspace{0.2em} 0 \hspace{0.2em}$ as the coefficient). The terms are in decreasing order of their degrees.

Step 3. Write k and the coefficients of the dividend in a row as shown below.

Step 4. Bring the leading coefficient down into the third row (leaving the second one).

Step 5. Multiply the term at the bottom (leading coefficient in the third row) by k and write it in the next column of the second row.

Step 6. Add the terms in the second column and write the result below.

Step 7. Repeat steps 5 and 6 with the remaining columns.

Once you reach the end, it’s time to interpret the result.

Step 8. The number in the last column is our remainder. The numbers before that give us the quotient polynomial.

Step 9. The quotient polynomial will be a polynomial with a degree one less than that of the dividend. Its coefficients are given by the numbers in columns — one through second-last. (Remember, the last column gives the remainder).

Factoring and Dividing

This method requires you to be comfortable with factoring polynomials. For this section, I am assuming you are.

In many cases, it is possible to divide one polynomial by another without having to use long division or synthetic division. This is especially true if the degree of the dividend is less than or equal to $\hspace{0.2em} 2 \hspace{0.2em}$ .

What you need to do in such cases is — factorize the dividend and see if you can cancel out any factors. Let me show how using a few examples.

Examples

Simplify.

\begin{align*} &(i) \hspace{1.1em} \frac{x^2 - 4}{x + 2} \\[1em] &(ii) \hspace{0.9em} \frac{x^2 - 6x + 5}{x - 1} \end{align*}

Solution ( $\hspace{0.2em} i \hspace{0.2em}$ )

Here’s an important algebraic identity that you might have come across.

a^2 - b^2 \hspace{0.25em} = \hspace{0.25em} (a + b)(a - b)

Now, we can use this identity to factorize the dividend (the top part).

\begin{align*} {\color{Red} x} ^2 - {\color{Teal} 4} \hspace{0.25em} &= \hspace{0.25em} {\color{Red} x} ^2 - {\color{Teal} 2} ^2 \\[1em] &= \hspace{0.25em} ( {\color{Red} x} + {\color{Teal} 2} )( {\color{Red} x} - {\color{Teal} 2} ) \end{align*}

And in the last step, we cancel out the common factors.

\begin{align*} \frac{x^2 - 4}{x + 2} \hspace{0.25em} &= \hspace{0.25em} \frac{\cancel{( {\color{Red} x + 2} )}(x - 2)}{\cancel{ {\color{Red} x + 2} }} \\[1em] & \hspace{0.25em} x - 2 \end{align*}

Solution ( $\hspace{0.2em} ii \hspace{0.2em}$ )

In this example, we have a quadratic trinomial on top. So we can factorize it using the unFOIL method.

x^2 - 6x + 5 \hspace{0.25em} = \hspace{0.25em} (x - 5)(x - 1)

And then just as in the last example, we cancel out the common factors.

\begin{align*} \frac{x^2 - 6x + 5}{x - 1} \hspace{0.25em} &= \hspace{0.25em} \frac{(x - 5)(\cancel{ {\color{Red} x - 1} })}{\cancel{ {\color{Red} x - 1} }} \\[1em] &= \hspace{0.25em} x - 5 \end{align*}

And with that, we come to the end of this tutorial on dividing polynomials. Until next time.