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Synthetic Division Calculator

Enter the dividend and divisor to perform synthetic division on a polynomial expression.

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Synthetic Division Calculator

Use this synthetic division calculator to divide polynomials by linear binomials. It provides step-by-step synthetic division to compute the quotient and remainder efficiently.

What Is Synthetic Division?

Synthetic division is a shortcut method for dividing a polynomial by a linear factor with a leading coefficient of 1. If the leading coefficient is not 1, you must first adjust the dividend. It gives both the quotient and remainder and is mainly used for divisors of the form (x + a) or (x - a).

Polynomials act as dividends
Linear factors of the form (ax + b) serve as divisors

Synthetic Division Formula:

\( \frac{P(x)}{(x-a)} = Q(x) + \frac{R}{(x-a)} \)

P(x): Dividend polynomial of any degree
(x-a): Linear divisor of degree 1
Q(x): Quotient polynomial
R: Remainder

How To Perform Synthetic Division

Write the coefficients of the dividend in descending order.
Find the zero of the linear factor (x - a = 0 → x = a).
Set up the synthetic division table.
Carry down the first coefficient.
Multiply the carried value by the divisor and write it under the next coefficient.
Repeat multiplication and addition for all coefficients.
The last number is the remainder; 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:

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

Why Choose Our Synthetic Division Calculator?

Fast & Efficient: Quickly divides polynomials without manual errors.
Step-by-Step Guide: Provides detailed steps for learning and verification.
Simple Interface: Easy to use; no advanced math skills required.

FAQ's

When Is Synthetic Division Applicable?

It works when the divisor is a linear factor of the form ax + b. Use this calculator for automatic step-by-step solutions.

Does Synthetic Division Work For All Polynomials?

No, it only works for dividing polynomials by linear expressions (x - c), where c is a constant.

Are There Any Limitations?

This calculator is designed specifically for synthetic division and cannot handle non-linear divisors or more complex algebraic divisions.

References

Lumen Learning - Synthetic Division

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