Polynomial Long Division Calculator

An online polynomial long division calculator allows the division of two polynomials to figure out the remainder and quotients. This calculator is utilized to perform long divisions within simple steps. 

What is a Long Divisions Polynomial?

“The process of dividing one polynomial by the other of the same or lower degree is known as long division polynomials”.

Algorithm for Division Polynomials:

This process implements the Euclidean division of polynomials. The division algorithm for polynomials says if p(x) and g(x) are the two polynomials in which g(x) ≠ 0. 

So, we can write the division of polynomials as: 

p(x) = q(x) × g(x) + r(x) 

Where, 

• p(x) is the dividend
• q(x) is the quotient.

Rules for Polynomial Divisions:

Usually, at the time of calculations, the general rule that is considered is “Divide, Multiply, Subtract, Bring Down, and Repeat”. 

1. The higher power is written first and then the lower power is written in descending order. 

2. Divide the higher term by the first term of the divisor and write the result above the line as the first term of the quotient. 

3. Multiply the entire divisor by the quotient put the divisor under the line and after that subtract it from the original polynomial 

4. Repeat the process till the degree of remainder is less than the divisor or there is no value left that is not dividable

5. The final leftover value is the remainder and the term that is above the line is the quotient.

Practical Example:

$$ \dfrac{4 x^{3} - 5 x^{2} + 11 x - 9}{x + 4} $$

Solution: 

Our online polynomial long division calculator will perform the long division of polynomials, with the steps shown. Hence, write the problem in the special format (missed terms are written with zero coefficients):

$$ \require{enclose} \begin{array}{rrrrrr} \\x + 4&\phantom{-} \enclose {longdiv}{\begin{array}{cccccc} 4x^3   & -   5x^2  & +  11x & - 9\end{array}}\end{array} $$

Step # 1:

Divide the leading term of the dividend by the leading term of the divisor: $$ \space \dfrac{4 x^{3}}{x} = 4 x^{2} $$

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

Multiply it by the divisor: $$ \space 4 x^{2}(x + 4) = 4 x^{3} + 16 x^{2} $$

Subtract the dividend from the obtained result: $$ \space (4 x^{3} - 5 x^{2} + 11 x - 9) - (4 x^{3} + 16 x^{2}) = - 21 x^{2} + 11 x - 9 $$

Step # 2:

Divide the leading term of the dividend by the leading term of the divisor: $$ \space \dfrac{- 21 x^{2}}{x} = - 21 x $$

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

Multiply it by the divisor: $$ \space - 21 x(x + 4) = - 21 x^{2} - 84 x $$

Subtract the dividend from the obtained result: $$ \space (4 x^{3} - 5 x^{2} + 11 x - 9) - (- 21 x^{2} - 84 x) = 95 x - 9 $$

Step # 3:

$$ \text {Divide the leading term of the dividend by the leading term of the divisor:} \space \dfrac{95 x}{x} = 95 $$

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

Multiply it by the divisor: $$ \space 95(x + 4) = 95 x + 380 $$

Subtract the dividend from the obtained result: $$ \space (4 x^{3} - 5 x^{2} + 11 x - 9) - (95 x + 380) = -389 $$

Result Table:

$$ \require{enclose}\begin{array}{rlc}
                \phantom{x + 4}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrrrr}
                  4 x^{2} & - 21 x & + 95&\end{array}&\\x + 4&\phantom{-}\enclose{longdiv}{\begin{array}{cccccc}4x^3   & -   5x^2  & +  11x & - 9\end{array}}\\&\begin{array}{rrrrrr}-\\\phantom{\enclose{longdiv}{}}
                    4 x^{3} & + 16 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 21 x^{2} & + 11 x & - 9                                      \\&-\\\phantom{\enclose{longdiv}{}}&- 21 x^{2} & - 84 x\\\hline\phantom{\enclose{longdiv}{}}&&95 x & - 9                                      \\&&-\\\phantom{\enclose{longdiv}{}}&&95 x & + 380\\\hline\phantom{\enclose{longdiv}{}}&&&-389                                    \\\\\phantom{\enclose{longdiv}{}}&&&\end{array}&\begin{array}{c}\\\phantom{}
                \end{array}\end{array} $$

Quotient:

$$ \text{So, the quotient is} \space {4 x^{2} - 21 x + 95} $$

Reminder:

$$ \text{and the remainder is} \space {-389} $$

Therefore the answer which is verified by the polynomial long division calculator is:

$$ \dfrac{4 x^{3} - 5 x^{2} + 11 x - 9}{x + 4} = {4 x^{2} - 21 x + 95+\dfrac{(-389)}{x + 4}} $$

Working of Polynomial Long Division Calculator:

The polynomial long division calculator allows you to find the quotient and the remainder of any polynomial functions fast and accurately! Providing the following values is required to make it work:

Input:

• Put the values of dividend and divisor 
• Tap on the “Calculate” button

Output:

Using our polynomial long division calculator, you will get the following results.

• Result Table 
• Quotient and remainder 
• Step-by-step solution

References:

From the source Wikipedia: Polynomial long division, Example, Pseudocode, Euclidean division, Applications. 

From the source Libretext: Dividing by a Polynomial, Polynomial functions example.