How to Do Long Division of Polynomials - A Complete Guide

Whether you're just starting to learn about the long division of polynomials in school, getting ready for a standardized math test, or looking to refresh your skills, you’re in the right place. 

Read on to find simple definitions, step-by-step instructions, solved examples, practice exercises, and answers to the most common questions about polynomial division.

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Let's Start with Definitions: What Are Polynomials and Long Division?

Long division of polynomials might sound like a mouthful, but don’t worry—it’s not as scary as it seems. 

Since it brings together two separate concepts, let’s break them down and understand each one before we tackle them together.

What Is a Polynomial?

A polynomial is an algebraic expression made up of variable terms like 3x² or 5x² and constant terms like 7 or -2, combined using addition and subtraction.

Each variable in a polynomial has a non-negative integer exponent like x³, x¹ or x⁰ but never x^-1 or x^{\(\frac{1}{2}\)}.

For instance: 

4x³ − 2x + 7 is a polynomial because it has:

• Variable terms: 4x³ and −2x

• Constant term: 7 

• Non-negative integer exponents: 3 (in 4x³) and 1 (in −2x) 

Polynomials can have different degrees, which is the highest power of the variable terms. 

For example, in 3x² + 5x − 2, the degree is 2 because the highest power of x is x².  This means that the term with x² has the largest exponent, which determines the degree of the polynomial.  

In 4x³ −7x² + 2x + 1, the degree is 3 because the highest power of x  is x³.

What Is Long Division?

At its core, division is counting how many of “these” fit into “that.” It’s like asking, “How many groups of 4 can I count inside 20?” 

Building on that, long division is a process, usually for bigger numbers, that breaks division down into simpler steps to follow. This makes it easier to work through problems step by step instead of all at once. 

The long division process consists of four key components: 

• Dividend: The number we are dividing. 

• Divisor: The number we are dividing by. 

• Quotient: The result or answer of the division. 

• Remainder: Any leftover amount if the division doesn't result in a whole number. 

Each long division task follows the same basic operations: 

1. Divide: Starting from the left, determine how many times the divisor fits into part or all of the dividend. 

2. Multiply: Multiply the divisor by the current quotient digit. 

3. Subtract: Subtract the result from the current portion of the dividend. 

4. Bring Down: Bring down the next digit from the dividend. 

5. Repeat: Continue the process until there is nothing left to bring down. 

How to Do Long Division of Polynomials 

The approach to diving polynomials is similar to dividing large numbers, but instead of numbers, we’re working with terms that include variables like x.

The steps, when applied to polynomial long division, become: 

1. Divide: Determine how many times the leading term of the divisor fits into the leading term of the dividend. 

2. Multiply: Multiply the entire divisor by this quotient term. 

3. Subtract: Subtract the result from the dividend to find the new remainder. 

4. Bring Down: Bring down the next term of the dividend. 

5. Repeat: Continue the process until there are no terms left to bring down. 

Now, to see how this works in action.  

We will divide \(\Large\frac{x^2 + 4x + 3}{x + 1}\) where x \(\neq\) −1.

Before we start dividing, we’ll write the dividend (x² + 4x + 3) under the division symbol and the divisor (x + 1) outside the division symbol to the left like so:

Now, we follow the steps: 

Step 1: Divide 

Look at the leading term of the divisor (x). Determine how many times it goes into the leading term of the dividend (x²). 

x² ÷ x = x

Write x above the division symbol. 

Step 2: Multiply & Subtract 

Write the product of the dividend (x + 1) and the quotient (x) below the like terms, lined up by degree. 

x × x + 1 = x² + x

Use parentheses and subtract the result from the first two terms of the dividend.

Let’s go term by term: 

• x² − x² = 0

• 4x − x = 3x

We write 3x below the line. 

Step 3: Bring Down 

We bring down the next term of the dividend (+3).

Step 4: Divide Again 

If the degree of the remainder is greater than or equal to the degree of the divisor, divide again.   

Determine how many times the leading term of the divisor (x + 1), goes into the leading term of the dividend (3x + 3).

3x ÷ x = 3

We write +3 above the division symbol, next to x.  

Step 5: Multiply & Subtract Again  

Write the product of x + 1 and 3 below the like terms, lined up by degree 

3 × (x + 1) = 3x + 3

Use parentheses and subtract it from 3x + 3. 

Let’s go term by term: 

• 3x − 3x = 0

• 3 − 3 = 0

Both terms cancel out. We write 0 below the line.

Since the remainder is 0, the division is complete.

Our final result is: \(\Large\frac{x^2 + 4x + 3}{x + 1}\) = x + 3

And that’s how we do long division of polynomials when there’s no remainder.  

 

How to Do Long Division of Polynomials with Remainders 

In some cases, when dividing polynomials, the division doesn’t work out evenly.  

This means we’ll have a remainder left over, just like when dividing regular numbers (e.g., 13 ÷ 4 = 3 with a remainder of 1). 

When this happens, we write the remainder as a rational expression—placing it over the divisor—so our final answer accurately represents the division. 

Now, let’s see this in action with an example! 

We’ll divide \(\Large\frac{x^2 − 5x + 7}{x − 1}\) where x \(\neq\) 1. 

Before we start the division, we place the dividend (x² − 5x + 7) under the division symbol and the divisor (x − 1) outside the division symbol to the left. 

Step 1: Divide 

Look at the leading term of the divisor, x, and determine how many times it goes into the leading term of the dividend, x². 

x² ÷ x = x

Write x above the division symbol as the first term of our quotient.

Step 2: Multiply & Subtract 

Write the product of x – 1 and x below the appropriate terms, lined up by degree.

x(x − 1) = x² − x

Use parentheses and subtract the result from the first two terms of the dividend. 

Let’s go term by term 

•  x² − x² = 0 

• −5x − (−x) = −5x + x = −4x 

Since x² terms cancel each other out, we only write −4x below. 

Step 3: Bring Down 

We bring down the remaining term of the dividend. 

Step 4: Divide Again  

If the degree of the remainder is greater than or equal to that of the divisor, divide again.   

Determine how many times x goes into –4x. 

−4x ÷ x = −4 

We write –4 above the division symbol, next to x. 

Step 5: Multiply & Subtract Again  

Write the product of x – 1 and –4 below the appropriate terms, lined up by degree.   

−4(x − 1) = −4x + 4

Use parenthesis and subtract the result from −4x + 7. 

Let’s go term by term: 

• −4x − (−4x) = −4x + 4x = 0

• 7 − 4 = 3

Write the result (3) below.

Here, x − 4 is our quotient and 3 is our remainder. Since the remainder, 3, has a lower degree (degree of 0) than the divisor x − 1 (degree of 1), we stop here. 

To express the remainder, we write it as a rational expression over the divisor: 

x − 4 + \(\Large\frac{3}{x − 1}\)

So, our final answer is: 

\(\Large\frac{x^2 − 5x + 7}{x − 1}\) = x − 4 + \(\Large\frac{3}{x − 1}\)

Long Division of Polynomials by Missing Terms 

Sometimes, a polynomial is missing one or more powers of the variable when written in standard form.  

When a term is missing, we can write an equivalent polynomial by adding placeholder terms with a coefficient of zero. 

This makes sure that each term lines up correctly in the long division process and helps avoid errors. 

Let’s go through an example to see exactly how we do it! 

We’ll divide \(\Large\frac{x^3 − 14x − 8}{x − 4}\) where x \(\neq\) 4.

Since the dividend x³ − 14x − 8 is missing an x²-term, we insert 0x². 

So, we write x³ + 0x² − 14x − 8 under the division symbol and x − 4 outside to the left like so: 

Now, we can divide, step by step. 

Step 1: Divide

Look at the leading term of the divisor, x, and determine how many times it goes into the leading term of the dividend, x³: 

x³ ÷ x = x²

Write x² above the division symbol, right over x³. 

Step 2: Multiply & Subtract 

Write the product of x – 4 and x² below the appropriate terms, lined up by degree.   

x²(x − 4) = x³ − 4x²

Use parentheses and subtract the result from the first two terms of the dividend. 

We subtract term by term: 

• x³ − x³ = 0

• 0x² − (−4x²) = 0 + 4x² = 4x²

Since x³ terms cancel each other out, we write 4x² as our result.

Step 3: Bring Down 

We can bring down the next term of the dividend, −14x.

This gives us the new expression: 

4x² − 14x

Step 4: Divide Again 

Determine how many times the leading term of the divisor (x) goes into 4x²:  

4x² ÷ x = 4x

Above the division symbol, add +4x as the second term of the quotient, to the right of x². 

Step 5: Multiply & Subtract Again 

Write the product of x – 4 and 4x below the appropriate terms, lined up by degree.   

4x(x − 4) = 4x² − 16x

Use parentheses and subtract this result from 4x² − 14x.

We can go term by term to avoid mistakes: 

• 4x² − 4x² = 0

• −14x − (−16x) = −14x + 16x = 2x

Since x² terms cancel each other out, we only write 2x as our result below.

Step 6: Bring Down 

We can bring down the next term of the dividend and place it next to 2x.

Step 7: Divide Again 

Determine how many times the leading term of the divisor (x) goes into −2x:  

2x ÷ x = 2

Write the result (+2) as the third term of the quotient, to the right of 4x. 

Step 8: Multiply & Subtract Again 

Write the product of x – 4 and 2 below the appropriate terms, lined up by degree.   

2(x − 4) = 2x − 8

Use parentheses and subtract this result from 2x − 8. 

Let’s go term by term: 

• 2x − 2x = 0

• −8 − (−8) = −8 + 8 = 0

Both terms cancel each other out, so we write 0 below. 

Since there’s no remainder, we stop dividing here. 

So, using long division we found:  

\(\Large\frac{x^2 − 14x − 8}{x − 4}\) = x² + 4x + 2

Long Division of Polynomials with a Non-Linear Divisor

So far, we’ve divided polynomials by linear divisors (divisors of the form ax+b, where the highest exponent is 1).  

But what if the divisor itself is a polynomial with a higher degree, like x² + 3x?  

We follow the same long division steps as before—divide, multiply, subtract, bring down, and repeat—until the remainder has a lower degree than the divisor. 

Let’s see how it’s done with an example: 

\(\Large\frac{2x^3 + x^2 − 4x + 16}{x^2 + 3x}\) where x \(\neq\) 0 or 3. 

Since the process mirrors what we did for linear divisors, we’ll show the steps visually to reinforce the concept.

Since the expression 11x + 16  has a lower degree than the divisor x² + 3x, we stop here. 

We write the remainder as a rational expression by placing it over the divisor: 

2x − 5 + \(\Large\frac{11x + 16}{x^2 + 3x}\)

Using long division, we found: 

\(\Large\frac{2x^3 + x^2 − 4x + 16}{x^2 + 3x}\) = 2x − 5 + \(\Large\frac{11x + 16}{x^2 + 3x}\)

 

More Solved Examples of Long Division of Polynomials 

Practice makes perfect!  

We've prepared a few more solved examples to show the long division of polynomials in action. 

 

Example 1: Long Division of Polynomials without Remainders 

We’ll divide \(\Large\frac{x^2 − 7x  + 12}{x − 3}\) where x \(\neq\) 3. 

Since we’re already familiar with the steps of polynomial long division, we’ll present the entire solution visually for clarity. 

So, using long division, we found \(\Large\frac{x^2 − 7x  + 12}{x − 3}\) = x − 4

Example 2: Long Division of Polynomials with Remainders 

To revisit the long division of polynomials with remainders, we’ll divide \(\Large\frac{x^2 + 3x + 5}{x + 2}\).

Here’s how the process unfolds.

Here, x + 1 is our quotient and 3 is our remainder. Since the remainder, 3, has a lower degree (degree of 0) than the divisor x + 2 (degree of 1), we stop here. 

We express the result and the remainder as: 

x + 1 + \(\Large\frac{3}{x + 2}\)

In the end, we found that \(\Large\frac{x^2 + 3x + 5}{x + 2}\) = x + 1 + \(\Large\frac{3}{x + 2}\). 

Example 3: Long Division of Polynomials by Missing Terms 

Now, to strengthen our understanding, we’ll practice another example of long division with a missing degree. 

We’ll divide \(\Large\frac{x^3 − 19x − 30}{x − 5}\) where x \(\neq\) 5. 

Since the dividend x³ − 19x − 30 is missing x² term, we insert 0x² as a placeholder. 

Next, we’ll go through the long division process and present the solution visually!

By applying long division to a polynomial with missing terms, we found: 

\(\Large\frac{x^3 − 19x − 30}{x − 5}\) = x² + 5x + 6. 

Example 4: Long Division of Polynomials with a Non-Linear Divisor 

 Lastly, we’ll practice long division of polynomials with a non-linear divisor. This time, we’ll divide: 

\(\Large\frac{x^3 + 6x^2 + 4x + 22}{x^2 + 3}\)

Now, let’s go through the solution visually! 

Since the remainder x + 4 has a lower degree than the divisor x² + 3, we stop here. 

We write the remainder as a fraction over the divisor: 

x + 6 + \(\Large\frac{x + 4}{x^2 + 3}\)

The final result is \(\Large\frac{x^3 + 6x^2 + 4x + 22}{x^2 + 3}\) = x + 6 + \(\Large\frac{x + 4}{x^2 + 3}\).

Practice Questions: Long Division of Polynomials   

Now it’s time to test your skills with long division of polynomials!

Try to solve these 4 expressions:  

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FAQs About the Long Division of Polynomials 

From the basics to the tricky bits, we’ve gathered the most common questions students have about dividing polynomials.   

 

1) What do I do when there’s no remainder? 

If there is no remainder (meaning the remainder is 0), the final answer is simply the quotient you obtained from the long division process. 

 

2) What if the degree of the divisor is greater than the degree of the dividend? 

Polynomial long division is typically used when the dividend has a degree equal to or greater than the divisor. 

If the divisor has a higher degree, then long division isn’t applied, but that doesn’t mean the expression is fully simplified. 

Instead, the expression remains as a rational expression, and you should check for common factors. 

If the numerator and denominator share a factor, the expression can still be simplified by factoring and reducing. 

 

3) How do I know when to stop dividing? 

You stop dividing when the degree of the remainder is less than the degree of the divisor.  

At this point, further division isn’t possible, and the remainder (if any) is written as a rational expression placed over the divisor.  

If the remainder is 0, the quotient is the final answer. 

 

4) Do I always need to write the remainder as a fraction? 

If you want an exact answer, yes. The remainder is written as a rational expression placed over the divisor.  

For instance, dividing x² + 5x + 4 by x − 2 might give a quotient of x + 7 with a remainder of 18, so the final answer is x + 7 + \(\Large\frac{18}{x − 2}\). 

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