Synthetic Division Calculator

The synthetic division calculator will calculate the quotient and remainder of the given divisor and dividend. The synthetic division solver makes it possible to find the answer only by using the coefficient of the polynomials.

What is Synthetic Division?

The division of polynomials is a special case of dividing a polynomial expression by a linear factor. It is used for determining the zeros of the polynomial by the division method. In the synthetic division method, the leading coefficient should equal 1. The synthetic division is normally  used to find the zero of the polynomial with a zeros calculator

Synthetic Division Formula

The formula of synthetic division is stated as follows.

P(x)/(x-c) = Q(x) + R/(x-c)

Where

P(x) = Dividend Polynomial of Any Order

(x-c) = Linear Factor of Degree “1”

Q(x) = Quotient 

R = Remainder

How to Divide Polynomials Using Synthetic Division?

You can do synthetic division manually but it’s a challenging task, The following steps are used by the divide using a polynomial synthetic division calculator with steps for the division process:

Step 1: Start the synthetic division

• To find the number to substitute it in the division box, we need to set the denominator polynomial as zero.
• If any term is missing, then write zero to fill in the missing term and write the numerator polynomial in descending order.

Step 2: Write Down the Coefficients

• Bring the leading coefficient straight down when the problem is set up perfectly.

Step 3: The Brought-Down Number

• Substitute the outcomes in the next column by multiplying the number in the division box with the brought-down number.

Step 4:Bring the Outcome Down

• By substituting two numbers, write the outcome at the bottom of the row.

Step 5:Continue to  the End

• Write the final results.
• The variables shall start with one power less than the denominator polynomial and go down with every term.

Step 6:Determine the Quotient and Remainder:

Write down the quotient and the remainder of the synthetic division

Example:

Let’s conduct the synthetic division with the dividend as 7x^3 + 4x + 8 and the divisor x + 2.

Given:

Dividend =  7x^3 + 4x + 8

Divsor =  x + 2

Solution:

\(frac {7x^3 + 4x + 8}{x + 2}\)
Coefficient of the numerator and denominator  polynomial of first-degree binomial expression are:

$$ 7, 4, 8 $$

$$ X + 2 = 0 $$

$$ X = −2.0 $$

The problem with the division format

$$\begin{array}{c|rrrrr}&x^{3}&x^{2}&x^{1}&x^{0}\\-2.0&7&0&4&8\\&&\\\hline&\end{array}$$

Carry down the leading coefficient to the bottom row

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&\\\hline&7\end{array} \)

$$ 7∗(−2.0) = −14 $$
\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&\\\hline&7&\end{array} \)

$$ 0 + (−14) = −14 $$
\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&\\\hline&7&-14&\end{array} \)

Multiply the obtained value by the zero of the denominators, and put the outcome into the next column
$$ −14 ∗ (−2.0) = 28 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&\\\hline&7&-14&\end{array} \)
Add down the column
$$ 4 + (28) = 32 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&\\\hline&7&-14&32&\end{array} \)
The division solver multiplies the obtained value by the zero of the denominators and puts the outcome into the next column
$$ 32 ∗ (−2.0 ) = −64 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&-64&\\\hline&7&-14&32&\end{array} \)

$$ 8 + (−64) = −56 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&-64&\\\hline&7&-14&32&-56&\end{array} \)

So, the quotient is \( 7x^2−14x+32 \), and the division remainder is −56 Therefore, the Answer is:

$$ \frac{7x^3 + 4x + 8} {x + 2} $$
$$ 7x^2 − 14x + 32 − \frac {56} {x + 2} $$

Once entering the polynomial of the degree one in the synthetic division solver, then able to gather the result of the division by the coefficient of the variable.

How Does Synthetic Calculator Work?

The calculator functions to perform synthetic division of polynomial coefficients that you need to enter in it. 

Let’s find out more about working!

Input:

• First, substitute the polynomials as dividend and divisor.
• Click on the “Calculate” button.

Output:

• The how-to synthetic division polynomials calculator finds the coefficients of the numerator and the zero of the denominator.
• It also provides the quotient and the remainder of polynomials.

Reference:

From the source of Wikipedia: Regular synthetic division

From the source of Lumen Learning: Two Polynomials, Use Synthetic Division to Divide