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Polynomial Long Division Calculator

Polynomial Long Division Calculator

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An online polynomial long division calculator will help you to perform the long division of a given dividend and divisor. You can find the remainder and quotient with a polynomial division calculator that provides detailed calculations for long polynomial division. You can learn and understand the whole concept of how to do long division with polynomials and much more.

What is Polynomial Long Division?

In algebra, long division of polynomials is an algorithm for dividing the polynomial, where a polynomial divide by the other polynomial of the same or lower degree. Therefore the generalized version of the familiar arithmetic method is called long division polynomials.
It can be done easily with the assistance of a dividing polynomials calculator because it separates the complex division problems into smaller ones.

How to do Polynomial Long Division?

The long division polynomials method can be performed by two different polynomials. This method is often used by the divide polynomials calculator to break down the complex form into the simplest form. However, we demonstrate the step-by-step solution for long division with polynomials in the given example.

Example:

Find the quotient and the reminder with long division, where the dividend is \( 2x^3 – 3x^2 + 13x – 5 \) and the divisor is x + 5.

Solution:

Missed terms are written with zero coefficients:
$$ \require {enclose} \begin {array} {rrrrrr} \\x + 5&\phantom {-} \enclose {longdiv} {\begin {array} {cccccc} 2x^3 & – 3x^2 & + 13x & – 5\end {array}} \end {array} $$

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:

Divide the leading term of the dividend by the leading term of the divisor: \( \space \dfrac{- 13 x^{2}} {x} = – 13 x \)
Multiply it by the divisor: \( \space – 13 x(x + 5) = – 13 x^{2} – 65 x \)
Subtract the dividend from the obtained result: \( \space (2 x^{3} – 3 x^{2} + 13 x – 5) – (- 13 x^{2} – 65 x) = 78 x – 5 \)

Step 3:

Divide the leading term of the dividend by the leading term of the divisor: \( \space \dfrac{78 x}{x} = 78 \)
Multiply it by the divisor: \( \space 78(x + 5) = 78 x + 390 \).
Subtract the dividend from the obtained result: \( \space (2 x^{3} – 3 x^{2} + 13 x – 5) – (78 x + 390) = -395 \)

Result Table:

You could try a polynomial long division calculator to attain the complete result table for quotient and remainder.
$$ \require {enclose} \begin{array} {rlc} \phantom{x + 5}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr} 2 x^{2} & – 13 x & + 78&\end{array}&\\x + 5&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}2x^3 & – 3x^2 & + 13x & – 5\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}} 2 x^{3} & + 10 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 13 x^{2} & + 13 x & – 5 \\&-\\\phantom{\enclose{longdiv}{}}&- 13 x^{2} & – 65 x\\\hline\phantom{\enclose{longdiv}{}}&&78 x & – 5 \\&&-\\\phantom{\enclose{longdiv}{}}&&78 x & + 390\\\hline\phantom{\enclose{longdiv}{}}&&&-395 \\\\\phantom{\enclose{longdiv}{}}&&&\end{array}&\begin{array}{c}\\\phantom{} \end{array}\end{array} $$
So, the quotient is \( 2x^2−13x+78 \), and the remainder is −395
Therefore, Answer is:
\( \dfrac{2 x^{3} – 3 x^{2} + 13 x – 5}{x + 5} = {2 x^{2} – 13 x + 78+\dfrac{(-395)}{x + 5}} \)

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

Types of Polynomial Division:

There are four different types of polynomial division:

  • Polynomial Division by another monomial
  • Division of a polynomial by nominal
  • Polynomial Division by binomial
  • Division of polynomial by another polynomial

The polynomial long division calculator can solve all types of polynomial division with a complete solution. Let’s discuss all types one by one:

Polynomial Division by Another Monomial:

For example an algebraic expression 40x^2 is divided by 10x then
$$ \frac {40 x^2} {10 x} $$
$$ \frac {5 * 2 * 2 * 2 * x * x} {2 * 5 * x} $$
Here, 5, 2, and x are common in both numerator and the denominator.
So,
$$ \frac {40 x^2} {10 x} = 4x $$

Division of a Polynomial by Nominal:

When a polynomial is divided by nominal, each term of a polynomial is separately divided by the monomial and the quotient of each division is substituted to obtain results. When you use the long division polynomials calculator for dividing the polynomial by a nominal it uses the long division method. You can do the division of polynomial by any nominal manually by different methods. Let’s check it with an example:

Example:

Divide the \( 24x^3 – 12xy + 9x \text{ by } 3x \).

Solution:

Given polynomial: \( 24x^3 – 12xy + 9x \)
It has three terms as:
\( 24x^3, 12xy, \text { and }, 9x \)
For division of expression with nominal, every term is divided by nominal separately as:
$$ (24x^3 / 3x) – (12xy / 3x) + (9x / 3x) $$
Hence,
$$ 8x^2 – 4y + 3 $$

Polynomial Division by Binomial:

The division of a polynomial by a binomial can be done easily because binomial is an expression with two terms. If we use a free online polynomial long division calculator for the division of polynomial by binomial, then it displays the polynomial in standard form. Now, use the long division method as bellow:

Example:

Divide the polynomial \( 3x^3 – 8x + 5 \) by x – 1.

Solution:

The dividend is \( 3x^3 – 8x + 5 \) and divisor is x – 1.
So,
Write the given expression in the special format:
$$ \require{enclose}\begin{array}{rrrrrr} \\x – 1&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}3x^3 & + 0x^2 & – 8x & + 5\end{array}}\end{array} $$

Step 1:

Divide the leading term of the dividend by the leading term of the divisor:
$$ \space \dfrac{3 x^{3}}{x} = 3 x^{2} $$
Multiply it by the divisor:
$$ \space 3 x^{2}(x – 1) = 3 x^{3} – 3 x^{2} $$
Subtract the dividend from the obtained result:
$$ \space (3 x^{3} – 8 x + 5) – (3 x^{3} – 3 x^{2}) = 3 x^{2} – 8 x + 5 $$

Step 2:

Divide the leading term of the dividend by the leading term of the divisor:
$$ \space \dfrac{3 x^{2}}{x} = 3 x $$
Multiply it by the divisor:
$$ \space 3 x(x – 1) = 3 x^{2} – 3 x $$
Subtract the dividend from the obtained result:
$$ \space (3 x^{3} – 8 x + 5) – (3 x^{2} – 3 x) = 5 – 5 x $$

Step 3:

Divide the leading term of the dividend by the leading term of the divisor:
$$ \space \dfrac{- 5 x}{x} = -5 $$
Multiply it by the divisor:
$$ \space -5(x – 1) = 5 – 5 x $$
Subtract the dividend from the obtained result:
$$ \space (3 x^{3} – 8 x + 5) – (5 – 5 x) = 0 $$

Result Table:

When using an online polynomial long division calculator, it shows the step-by-step calculations as:
$$ \require{enclose}\begin{array}{rlc} \phantom{x – 1}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr} 3 x^{2} & + 3 x & – 5&\end{array}&\\x – 1&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}3x^3 & + 0x^2 & – 8x & + 5\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}} 3 x^{3} & – 3 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&3 x^{2} & – 8 x & + 5 \\&-\\\phantom{\enclose{longdiv}{}}&3 x^{2} & – 3 x\\\hline\phantom{\enclose{longdiv}{}}&&5 & – 5 x \\&&-\\\phantom{\enclose{longdiv}{}}&&5 & – 5 x\\\hline\phantom{\enclose{longdiv}{}}&&&0 \\\\\phantom{\enclose{longdiv}{}}&&&\end{array}&\begin{array}{c}\\\phantom{} \end{array}\end{array} $$
So, the quotient is \( \space{3 x^{2} + 3 x – 5} \), and the remainder is 0
Therefore, Answer is:
$$ \dfrac{3 x^{3} – 8 x + 5}{x – 1} = {3 x^{2} + 3 x – 5} $$

However, an online Remainder Theorem Calculator allows you to determine the remainder of given polynomial expressions by remainder theorem.

Division of Polynomial by Another Polynomial:

For dividing polynomials long division with another polynomial, write the polynomial in standard form and use the long division method. Let us take an example as:

Example:

Perform long division of polynomials where the dividend is 3x^3 + x^2 + 2x + 5 and divisor is x^2 + 2x + 1.

Solution:

When you use polynomial long division calculator it display the problem statement in the special format that missed the terms with zero coefficients:
$$ \require{enclose}\begin{array}{rrrrrr} \\x^{2} + 2 x + 1&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}3x^3x^2 & + 2x & + 5\end{array}}\end{array} $$

Step 1:

Divide the leading term of the dividend by the leading term of the divisor:
$$ \space \dfrac{3 x^{3}}{x^{2}} = 3 x $$
Multiply it by the divisor:
$$ \space 3 x(x^{2} + 2 x + 1) = 3 x^{3} + 6 x^{2} + 3 x $$
Subtract the dividend from the obtained result:
$$ \space (3 x^{3} + x^{2} + 2 x + 5) – (3 x^{3} + 6 x^{2} + 3 x) = – 5 x^{2} – x + 5 $$

Step 2:

Divide the leading term of the dividend by the leading term of the divisor:
$$ \space \dfrac{- 5 x^{2}}{x^{2}} = -5 $$

Multiply it by the divisor:

$$ \space -5(x^{2} + 2 x + 1) = – 5 x^{2} – 10 x – 5 $$
Subtract the dividend from the obtained result:

$$ \space (3 x^{3} + x^{2} + 2 x + 5) – (- 5 x^{2} – 10 x – 5) = 9 x + 10 $$

Result Table:

$$ \require{enclose}\begin{array}{rlc} \phantom{x^{2} + 2 x + 1}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr} 3 x & – 5&\end{array}&\\x^{2} + 2 x + 1&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}3x^3x^2 & + 2x & + 5\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}} 3 x^{3} & + 6 x^{2} & + 3 x\\\hline\phantom{\enclose{longdiv}{}}&- 5 x^{2} & – x & + 5 \\&-\\\phantom{\enclose{longdiv}{}}&- 5 x^{2} & – 10 x & – 5\\\hline\phantom{\enclose{longdiv}{}}&&9 x & + 10 \\\\\phantom{\enclose{longdiv}{}}&&\end{array}&\begin{array}{c}\\\phantom{} \end{array}\end{array} $$
So, the quotient is 3x−5, and the remainder is 9x+10
Therefore, Answer is:
$$ \dfrac{3 x^{3} + x^{2} + 2 x + 5}{x^{2} + 2 x + 1} = {3 x – 5+\dfrac{(9 x + 10)}{x^{2} + 2 x + 1}} $$

How does Polynomial Long Division Calculator Works?

An online long division of polynomials calculator provides the division of two polynomials by following these steps:

Input:

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

Output:

  • The dividing polynomials calculator first displays the special format of the given values.
  • Then, it provides the result table, quotient, and remainder for given polynomials.

FAQ:

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.

Conclusion:

Use this online polynomial long division calculator for the long division of polynomials with detailed solutions. Engineers mostly use polynomials to model the path of the roller coasters. So, the polynomial division calculator precisely provides the result table with complete calculations for given values.

Reference: 

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

From the source of Purple Math: Long Polynomial Division, Division of Polynomial, Division of a monomial by another monomial, Division of a polynomial by a monomial.

From the source of Lumen Learning: Division Algorithm, Division of Polynomials, Rational Root Theorem, Remainder Theorem, Synthetic division, Rational Roots and Dividing Polynomials.