Polynomial Long Division Calculator

The polynomial long division calculator divides two polynomial expressions to find the quotient and remainder. The polynomial division with steps provides the user with a detailed insight into the long polynomial division.

What Is A Polynomial Long Division?

In algebra, the long division of polynomials is an algorithm for dividing the polynomial, where a polynomial is divided by another polynomial of the same or lower degree.

It can be done easily by dividing polynomials with steps because it separates complex division problems into smaller ones.

How To Do Polynomial Long Division?

Let us resolve an example to clarify the long division technique with polynomials!q

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:

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

Step 1:

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:

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

Step 2:

Repeating the steps again:

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

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

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

Step 3:

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

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

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

Result Table:

$$ \require {enclose} \begin{array} {rlc} \phantom{x + 5}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr} 2 x^{2} & – 13 x & + 78&\end{array}&\\x + 5&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}2x^3 & – 3x^2 & + 13x & – 5\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}} 2 x^{3} & + 10 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 13 x^{2} & + 13 x & – 5 \\&-\\\phantom{\enclose{longdiv}{}}&- 13 x^{2} & – 65 x\\\hline\phantom{\enclose{longdiv}{}}&&78 x & – 5 \\&&-\\\phantom{\enclose{longdiv}{}}&&78 x & + 390\\\hline\phantom{\enclose{longdiv}{}}&&&-395 \\\\\phantom{\enclose{longdiv}{}}&&&\end{array}&\begin{array}{c}\\\phantom{} \end{array}\end{array} $$
So, the quotient is \( 2x^2−13x+78 \), and the remainder is −395

Therefore, the Answer is:

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

Try a polynomial long division with remainders to attain the complete result table for quotient and remainder.

However, an online Synthetic division to find zeros will allow you to determine the reminder and quotient of polynomials using the synthetic division method.

Working of Calculator:

Using the online tool with a solution is very easy. It provides the division of two polynomials by following these steps:
Input:

• First, enter dividend and divisor in the given fields.
• Tap “Calculate”

Output:

• Then, it provides the result table, quotient, and remainder for given polynomials.

FAQs:

What Is Remainder And Quotient For The Polynomial Division xy / x + y?

The quotient is y and the remainder is \( -y^2 \) for the given polynomial expression xy / x + y.

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. And using these long-division polynomials can even speed up the calculations without trouble.

Reference: 

From the source of Wikipedia: Polynomial long and short division, Pseudocode, Euclidean division, Factoring polynomials, Finding tangents to polynomial functions.