Synthetic Division Calculator

The synthetic division calculator helps you to find the quotient and remainder of polynomials by using the synthetic division method. Also, find the coefficients of numerators and zeros of roots of polynomials by using this synthetic substitution calculator. 

What Is The Synthetic Division of Polynomials? 

“The Synthetic division is the shorthand method of dividing the polynomials when the divisor is a linear factor”.

It is generally used to determine the zeros of polynomials in which the divisor is in the form of (x ± n) where n indicates the whole number. 

Root Principle of Synthetic Division:

Get the synthetic divisions of the polynomial either by the leading coefficients should be one or by the linear expressions. The root principle to discover this division is:

“Bring down, Multiply and add, Multiply and add, Multiply and add, and so on”.

Keep in an account that there are two possibilities of synthetic method that are as follows:

• The leading coefficient must be equal to one 
• The Divisor of the given equation is also equal to one 

How to Calculate the Synthetic Division?

The divisions of polynomials can be done manually but it's a difficult task. By using the synthetic division of polynomials calculator this process can become easy for us. To divide using synthetic division calculator with steps look at the example below:

Example:

• Dividend is 4x^3 + 2x^2 + x + 8
• Divisor x + 2

Solution: 

$$ \dfrac{4 x^{3} + 2 x^{2} + x + 8}{x + 2} $$

Coefficients of the numerator polynomial

$$ 4, 2, 1, 8 $$

Find the zeros of the denominator

$$ x + 2 = 0 $$

$$ x = -2.0 $$

Write down the problem in synthetic division format

$$ \begin{array}{c|rrrrr}& x^{3}&x^{2}&x^{1}&x^{0} \\-2.0& 4&2&1&8 \\&&\\\hline&\end{array} $$

Carry down the leading coefficient to the bottom row

$$ \begin{array}{c|rrrrr}-2.0& 4&2&1&8 \\&&\\\hline&4\end{array} $$

Now, by the synthetic long division calculator multiply the obtained value by the zero of the denominator, and put the outcome into the next column

$$ 4 * (-2.0) = -8 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&\\\hline&4&\end{array} $$

Add down the column

$$ 2 + (-8) = -6 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&\\\hline&4&-6&\end{array} $$

Hence, by using the synthetic substitution calculator multiply the obtained value by the zero of the denominator, and put the outcome into the next column

$$ -6 * (-2.0) = 12 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&12&\\\hline&4&-6&\end{array} $$

Add down the column

$$ 1 + (12) = 13 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&12&\\\hline&4&-6&13&\end{array} $$

Multiply the obtained value by the zero of the denominator, and put the outcome into the next column

$$ 13 * (-2.0) = -26 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&12&-26&\\\hline&4&-6&13&\end{array} $$

Add down the column

$$ 8 + (-26) = -18 $$

$$ \begin{array}{c|rrrrr}-2.0&4&2&1&8\\&&-8&12&-26&\\\hline&4&-6&13&-18&\end{array} $$

$$ \text{So, the quotient is} \space \color{#39B54A}{4 x^{2} - 6 x + 13}, \space \text{and the remainder is} \space \color{#39B54A}{-18} $$

Therefore, the Answer is:

$$ \dfrac{4 x^{3} + 2 x^{2} + x + 8}{x + 2} = \color{#39B54A}{4 x^{2} - 6 x + 13 - \dfrac{18}{x + 2} } $$

Working of Synthetic Division Calculator:

To clarify the concept of how to divide polynomials using synthetic division method, the synthetic division solver is designed accurately! It functions only if you provide the following values:

Input:

• Dividend equation that changes the polynomial
• Put the Divisor like (ax ± b) 
• Tap “Calculate”

Output:

• Zeros of denominators 
• Coefficients of numerators 
• Remainder and quotients of polynomials 
• Steps in the form of synthetic division tables 

FAQs:

Can Be a Long Division Method Is Used Instead of Synthetic Division?

Synthetic division is the process of dividing the polynomials. If the polynomials have a degree 1 then it works well and if there is a higher degree that doesn’t lead to coefficients then a long division process can be used. 

References:

From the source Wikipedia: Synthetic division, Expanded synthetic division.

From the source Lumen Learning: Synthetic Division, Divide a Second-Degree Polynomials, Divide a Third-Degree Polynomial, Divide Fourth-Degree Polynomial, Application Problem.