Polynomial Long Division Calculator

Our polynomial long division calculator allows you to divide any polynomial by another, providing the quotient, remainder, and a detailed, step-by-step explanation. The polynomial division with steps provides the user with a detailed insight into the long polynomial division. Use it to check homework, simplify rational expressions, or solve complex polynomial problems effortlessly!

What is Polynomial Long Division?

Polynomial long division is a method in algebra used to divide one polynomial by another, breaking down complex expressions into simpler components for easier calculation and understanding.

This method is especially useful for:

Calculating the quotient and remainder of polynomials
Simplifying rational expressions
Analyzing the behavior of rational functions

How to Use this Polynomial Long Division Calculator?

Enter the dividend and divisor polynomials in the input fields
Click the CALCULATE button
View the quotient, remainder, and complete step-by-step division

Manual Steps for Polynomial Long Division

Follow these standard steps to divide polynomials manually:

Step 1: Arrange Polynomials

Write all polynomial terms in descending order of powers. Include missing terms with zero coefficients (Example: x³ + 2x + 1 → x³ + 0x² + 2x + 1).

Step 2: Divide Leading Terms

Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the divisor by the term obtained in Step 2 and subtract from the dividend to get the new remainder.

Step 4: Bring Down the Next Term

Bring down the next term from the dividend to form a new expression for the next division step.

Step 5: Repeat the Process

Continue dividing, multiplying, and subtracting until the remainder's degree is lower than the divisor's degree.

Step 6: Write the Final Answer

Record the quotient and any leftover remainder.

Example:

Divide 2x³ - 3x² + 13x - 5 by x + 5

Solution:

Write the problem in the special format (missed terms are written with zero coefficients)

\(\require{enclose}\begin{array}{rrrrrr} \\ x + 5&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}2x^3 & - 3x^2 & + 13x & - 5\end{array}}\end{array}\)

Multiply it by the divisor 1

Divide the leading term of the dividend by the leading term of the divisor: 2x³ ÷ x = 2x²

Write down the calculated result in the upper part of the table.

2x²(x + 5) = 2x³ + 10x²

Subtract the dividend from the obtained result:

(2x³ - 3x² + 13x - 5) - (2x³ + 10x²) = -13x² + 13x - 5

Multiply it by the divisor 2

Divide the leading term of the dividend by the leading term of the divisor:

-13x² ÷ x = -13x

Write down the calculated result in the upper part of the table.

-13x(x + 5) = -13x² - 65x

Subtract the dividend from the obtained result:

(-13x² + 13x - 5) - (-13x² - 65x) = 78x - 5

Multiply it by the divisor 3

Divide the leading term of the dividend by the leading term of the divisor:

78x ÷ x = 78

Write down the calculated result in the upper part of the table.

78(x + 5) = 78x + 390

Subtract the dividend from the obtained result:

(2x³ - 3x² + 13x - 5) - (78x + 390) = -395

Mathematically, the polynomial long division is:

\(\ \require{enclose}\begin{array}{rlc} \phantom{ x + 5 }&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr} 2 x^{2} & - 13 x & + 78&\end{array}&\\x + 5&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}2x^3 & - 3x^2 & + 13x & - 5\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}} 2 x^{3} & + 10 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 13 x^{2} & + 13 x & - 5 \\&-\\\phantom{\enclose{longdiv}{}}&- 13 x^{2} & - 65 x\\\hline\phantom{\enclose{longdiv}{}}&&78 x & - 5 \\&&-\\\phantom{\enclose{longdiv}{}}&&78 x & + 390\\\hline\phantom{\enclose{longdiv}{}}&&&-395 \\\\\phantom{\enclose{longdiv}{}}&&&78 x & + 390\end{array}&\begin{array}{c}\\\phantom{ -395 } \end{array}\end{array}\)

Final Answer:

\(\displaystyle 2x^3 - 3x^2 + 13x - 5 \div (x + 5) = 2x^2 - 13x + 78 + \left( \frac{-395}{x + 5} \right)\)

Polynomial Long Division vs Synthetic Division

Polynomial Long Division: Works for any polynomial divisor; detailed step-by-step method.
Synthetic Division: Shortcut method for linear divisors of the form x - c; faster but limited.

When to Use Long Division:

Divisor is not linear (degree > 1)
Leading coefficient is not 1
Need a detailed, step-by-step explanation

When to Use Synthetic Division:

Divisor is linear (x - c)
Quickly find remainder, factors, or evaluate polynomials

Common Mistakes & Tips

Incorrect Term Order: Always arrange polynomials from highest to lowest degree.
Skipping Zero Terms: Include missing terms with zero coefficients to maintain alignment.
Sign Errors: Be careful with subtraction of polynomials.
Stopping Too Early: Continue until the remainder has a smaller degree than the divisor.

Why Use Our Polynomial Long Division Calculator?

Supports polynomials of any degree
Step-by-step solutions for clear learning
Reduces errors with signs and missing terms
Fast and accurate results
Free and accessible on all devices
Downloadable results as PDF

FAQs

What is the Remainder Theorem?

If a polynomial f(x) is divided by x - c, the remainder is f(c). This allows quick remainder calculation without full division.

Can I divide by a polynomial of higher degree?

Yes, the quotient will be zero and the dividend itself becomes the remainder.

Why is the remainder’s degree less than the divisor’s degree?

The division stops when the remainder’s degree is smaller than the divisor’s degree, ensuring a valid quotient and remainder.

What Is The Best And Easiest Way To Divide The Long Polynomials?

The long division polynomials method is the best way to divide long polynomials. And using these long-division polynomials can even speed up the calculations without trouble. Polynomial long division is systematic, reliable, and provides clear steps for accurate results.

References

Wikipedia: Polynomial long division

LibreTexts: Using Long Division to Divide Polynomials

CK-12 Flexbooks: Long Division and Synthetic Division