Synthetic Division Calculator

Synthetic Division Calculator

The calculator uses the synthetic division method to divide a polynomial by a binomial with a step-by-step solution.

This Synthetic Division Calculator divides a polynomial (dividend) by a binomial (divisor) of the form (ax + b) using the synthetic division method. It provides step-by-step synthetic division to compute the solution efficiently. Simply enter the dividend polynomial and linear divisor to calculate the quotient and remainder using synthetic division.

What Is Synthetic Division?

Synthetic division is a shortcut method for dividing a polynomial of higher degree by a linear divisor (x - c) with a leading coefficient of 1. It simplifies the division process and helps to determine the quotient and remainder.

Synthetic division is used when the divisor is a linear polynomial (degree 1), as shown below:

• Can be solved using synthetic division:

x² + 3x + 2

x + 1

Here, the divisor x+1 has degree 1, so synthetic division can be used.

• Cannot be solved using synthetic division:

x³ + 2x² + x + 1

x² + 1

Here, the divisor x² + 1 has degree 2, so basic synthetic division cannot be used.

Synthetic Division Formula:

The synthetic division of polynomials formula is:

P(x) = (x − c) Q(x) + R

Where:

• P(x) = The dividend (the polynomial you are dividing)

• (x − c) = The divisor (the binomial you are dividing by)

• Q(x) = The quotient (the result of the division)

• R = The remainder (a constant value when dividing by a linear binomial)

How to Use the Synthetic Division Calculator?

To use this polynomial synthetic division calculator, you have to follow these steps:

Step 1: Add your dividend (polynomial) into the input field

Step 2: Add your divisor (binomial) in the form of ax + b

Step 3: Click the “CALCULATE” button to get the quotient, remainder, and step-by-step solution.

Optional: You can also check the “Equation Preview” to ensure your polynomial expression and divisor are in the correct format.

How to Do Synthetic Division of Polynomials?

To perform synthetic division in a sequence, follow the steps below:

Write coefficients of the dividend: Write down all the coefficients of the dividend polynomial in descending order, and also add a placeholder for the missing term.
Find the Zero of linear factor: Now equate the linear divisor to zero and solve for “x” (e.g., x - a = 0 → x = a) to determine the value of “a”.
Set up the Synthetic Division Table: Enter the coefficients of the dividend and the divisor into the synthetic division table.
Carry Down the First Coefficient: Now bring the leading coefficient straight down below the horizontal line.
Multiply and Add Across Columns: Multiply the bottom value by the “a”, place it under the next coefficient, and add them together.
Repeat for Remaining Coefficients: Repeat the multiplication process across all remaining coefficient columns of the polynomial dividend until you reach the end.
Interpret Quotient and Remainder: The last number is the remainder of our synthetic division, and the remaining numbers form the quotient.

Example

Divide the following polynomial using synthetic division:

Dividend: \( 2x^3 - 5x^2 + 3x - 7 \)

Divisor: \( x - 2 \)

Step-by-Step Solution:

Step 1: Write coefficients of the dividend

2, -5, 3, -7

Step 2: Find zero of linear factor

x - 2 = 0 → x = 2

Step 3: Set up synthetic division table

\(\begin{array}{c|rrrr} 2 & 2 & -5 & 3 & -7 \\ & & & & \\ \hline & & & & \end{array}\)

Step 4: Carry down the first coefficient

\(\begin{array}{c|rrrr} 2 & 2 & -5 & 3 & -7 \\ & & & & \\ \hline & 2 & & & \end{array}\)

Step 5: Multiply and add

2 × 2 = 4 → -5 + 4 = -1

\(\begin{array}{c|rrrr} 2 & 2 & -5 & 3 & -7 \\ & & 4 & & \\ \hline & 2 & -1 & & \end{array}\)

Step 6: Repeat until the last coefficient

\(\begin{array}{c|rrrr} 2 & 2 & -5 & 3 & -7 \\ & & 4 & -2 & 2 \\ \hline & 2 & -1 & 1 & -5 \end{array}\)

Step 7: Interpret the result

Quotient: \( 2x^2 - x + 1 \), Remainder: -5

Final Answer:

\( \frac{2x^3 - 5x^2 + 3x - 7}{x - 2} = 2x^2 - x + 1 - \frac{5}{x - 2} \)

Synthetic division is a fast way to divide polynomials by simple binomials like (x - c). This free synthetic division calculator online quickly finds quotients, checks roots, and verifies your manual calculations step by step.

Frequently Asked Questions:

How to handle a divisor like (x + c) in synthetic division?

In case your divisor is in the form (x + c), equate it with zero to find the value used for synthetic division: x + c = 0, which means x = -c. The “-c” is the divisor value outside the synthetic division box.

How to Interpret the Remainder in Polynomial Division?

A remainder gives you the exact numerical value of the original polynomial when you substitute the root of the divisor in place of the variable. If the remainder is zero, then it represents that the divisor is an exact factor of the polynomial.

Can Synthetic Division Be Used to Evaluate a Function?

Yes, synthetic division evaluates a polynomial function P(x) at a specific value x = c. The remainder obtained after dividing the polynomial by (x - c) is the exact functional value of P(c).

Can I use Synthetic Division for a Divisor Like x² + 1?

No. Synthetic Division is only used with linear divisors of the form (x − c). Since x² + 1 is not a linear divisor, use polynomial long division instead.

What If the Remainder Is Zero?

If the remainder is 0, the divisor divides the expression exactly. This means (x−c) is a factor, and c is a root (or zero) of the equation P(x)=0.

Can I Use Synthetic Division for Quadratic Divisors?

No. Standard synthetic division is designed for dividing polynomials by linear divisors of the form x−c. For quadratic divisors, such as x^2+2x+1, polynomial long division is generally used instead.`

References:

MathBitsNotebook - Polynomial Synthetic Division
Study.com - Synthetic Division of Polynomials