Polynomial Long Division Calculator

Polynomial Long Division Calculator:

This polynomial long division calculator divides one polynomial by another and shows the complete quotient, remainder, and step-by-step calculations. Whether you are a student, teacher, or engineer, our calculator helps you solve complex algebraic divisions instantly and accurately. Use it to simplify rational expressions, verify the homework, and complete the algebra assignments with ease!

What is Polynomial Long Division?

In algebra, polynomial long division is a step-by-step method for dividing a polynomial by another polynomial of the same or lower degree. This process breaks the complex polynomial expressions into smaller and more manageable parts. 

It’s a very helpful technique for:

• Finding the quotient and remainder of polynomials
• Simplifying rational expressions
• Understanding the behavior of rational functions

How to Use the Polynomial Long Division Calculator?

For the accurate working of the polynomial long division Calculator, follow these steps:

1. Enter the dividend and the divisor in the given fields
2. Click on the “CALCULATE” button
3. View the quotient, remainder, and step-by-step polynomial long division

How to Divide Polynomials using the Long Division Method?

Step-by-Step Process of Polynomial Long Division:

• Step 1: Arrange the polynomials

  • Note down the polynomial and arrange it in descending order of power. If you see there is a missing term, then add this term with a zero coefficient. (for example, write x³+2x+1x³ as x³+0x²+2x+1x³

• Step 2: Divide the Leading Terms 

  • Now 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 obtained term of the quotient. After that, subtract the result from the dividend to get the new remainder

• Step 4: Bring Down the Next Term

  • Form a new expression by bringing down the next term 

• Step 5: Repeat the Process

  • Keep dividing, multiplying, and subtracting until the degree of the dividend becomes lower than the degree of the divisor

• Step 6: Write Down the Final Result

  • Note down the quotient and the remainder (leftover, if any)

Solved Example of Polynomial Long Division:

Let us resolve an example of dividing polynomials to clarify the long division technique with polynomials! Find the quotient and the remainder with long division, where the dividend is 2x^3 - 3x^2 + 13x - 5 and the divisor is x + 5.

Solution:

\[ \begin{array}{r|l} x + 5 & 2x^3 - 3x^2 + 13x - 5 \\ \hline & 2x^2 - 10x + 78 \\ \end{array} \]

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

\( \space \dfrac{2 x^{3}}{x} = 2 x^{2} \)

Multiply it by the divisor:

\( \space 2 x^{2} (x + 5) = 2 x^{3} + 10 x^{2} \)

Subtract the dividend from the obtained result:

\( \space (2 x^{3} - 3 x^{2} + 13 x - 5) - (2 x^{3} + 10 x^{2}) = - 13 x^{2} + 13 x – 5 \)

Repeat the steps:

\( \space \dfrac{- 13 x^{2}} {x} = - 13 x \)

\( \space - 13 x(x + 5) = - 13 x^{2} - 65 x \)

\( \space (2 x^{3} - 3 x^{2} + 13 x - 5) - (- 13 x^{2} - 65 x) = 78 x - 5 \)

Write down the final result:

\( \space \dfrac{78 x}{x} = 78 \)

\( \space 78(x + 5) = 78 x + 390 \).

\( \space (2 x^{3} - 3 x^{2} + 13 x - 5) - (78 x + 390) = -395 \)

Result Table:

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

So, the quotient is \( 2x^2−13x+78 \), and the remainder is -395

Therefore, the Answer is:

\( \dfrac{2 x^{3} - 3 x^{2} + 13 x - 5}{x + 5} = {2 x^{2} - 13 x + 78+\dfrac{(-395)}{x + 5}} \)

That's how you can perform long division on a polynomial manually, but to save time and eliminate calculation errors, use our polynomial long division calculator. It quickly provides the quotient, remainder, and division steps for any given polynomial. 

When and Why to Use Polynomial Long Division vs Synthetic Division:

What is the difference between Polynomial Long Division and Synthetic Division?

• Polynomial Long Division: It is the method used to divide a polynomial by another polynomial
• Synthetic Division: This is a shortcut for performing division. It only works when the divisor is of the form “x-c”, where “c” is the constant. It is fast but less flexible as compared to the polynomial long division 

When to Use Long Division:

Use polynomial long division when:

1. The divisor is not linear (degree > 1)
2. Leading coefficient is not 1 (e.g., 2x+3 or x²+x+1)
3. Need a step-by-step explanation to understand the division process

When to Use Synthetic Division:

Use synthetic division when:

1. You have a linear divisor in the form of x-c
2. You want to quickly get the remainder, factors, or evaluate polynomials

For a quick, step-by-step solution, try our synthetic division calculator. It makes solving linear divisor problems accurate and easy.

✅Tip: When you see that the divisor is linear, use the synthetic division method. Otherwise, use the long division polynomials method.

Common Mistakes & Tips:

Common errors when performing polynomial long division:

• Forget Arranging Polynomials in descending Order: 

  • You must arrange the terms from the highest to the lowest order to prevent missing the steps. Missing the steps in the division process can cause errors 

• Skipping zero Coefficient Terms: 

  • If any term is missing (e.g., no x²) in the polynomial expression, then add it with a zero to maintain the proper alignment of the expression

• Sign Errors During Subtraction:

  • Pay attention to the negative signs and perform the subtraction with care

• Stopping Too Early:

  • Don't stop too early, keep performing the division until the remainder gets a lower degree than the divisor

Helpful Tips for Performing the Polynomial Long Division:

1. When you are performing the polynomial long division then you must write each step clearly to avoid confusion
2. Once you are done with the division process, verify your subtraction and alignment
3. Practice with different types of divisors to build confidence
4. Use the polynomial division calculator to verify the results and understand the process step-by-step

Why Use Our Polynomial Long Division Calculator?

This polynomial long division calculator provides an accurate, instant solution and an easy-to-understand explanation of the process. Here's why it stands out:

• Supports Any Degree of Polynomial:

  • You can use our calculator for dividing polynomials of any degree. It is not limited to only linear divisors

• Step-by-Step Solution:

  • Each step of the long division is clearly displayed with detailed explanations, making it easy to follow and learn

• Error Free Calculation:

  • Using our calculator prevents you from making frustrating sign mistakes or forgetting terms

• Speed & Accuracy:

  • Stop wasting time on the manual calculations when you can do that in just seconds. Simply input your dividend and divisor to get the 100% accurate division

• Free of Cost:

  • Use our free polynomial division calculator online with no usage limitations or restrictions. You can use it as many times as you want

• Device Friendly: 

  • Access the calculator on any smart device you have, PC, tablet, mobile phone, etc. Just access your browser, search for the calculator, and use it

• Downloadable Results:

  • After performing the calculation, you can download the results in the form of a PDF with just a single click on the download icon

FAQ’s:

What is the Remainder Theorem?

The polynomial division remainder theorem explained that if a polynomial f(x) is divided by a linear factor (x-c), then the obtained remainder is equal to f(c). It lets you easily find the remainder without requiring you to perform the full polynomial long division. 

Can I Divide by a Polynomial of Higher Degree?

Yes, you can perform the division using a polynomial of higher degree. In this case, the quotient is zero, and the dividend becomes the remainder.

Why is the Remainder Degree Less than the Divisor Degree?

As we know, the polynomial long division process continues until the degree of the remainder is less than the degree of the divisor. If the degree of remainder is higher than the degree of the divisor, then it will mean the division is not complete, and you will have to continue the division process. It makes sure that the remainder and quotient are correctly defined.

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

The long division polynomials method is the best way to divide two long polynomials. It provides a systematic, step-by-step approach, which makes it easy to handle complex polynomials. 

References: 

1. From the source of Wikipedia: Polynomial long and short division, Pseudocode, Euclidean division, Factoring polynomials, Finding tangents to polynomial functions.
2. From the source of math.libretexts.org: Using Long Division to Divide Polynomials.
3. From the source of flexbooks.ck12.org: Long Division and Synthetic Division.