Question 1

What is synthetic division?

Accepted Answer

Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x - c). Instead of the full long division process, you work only with the coefficients, making it faster and less error-prone. It works for any polynomial degree divided by a linear term.

Question 2

When can I use synthetic division?

Accepted Answer

Synthetic division only works when the divisor is a linear expression of the form (x - c), where the coefficient of x is 1. For divisors like (2x - 3), you would need to factor out the 2 first or use polynomial long division instead.

Question 3

What is the Remainder Theorem?

Accepted Answer

The Remainder Theorem states that when you divide polynomial f(x) by (x - c), the remainder equals f(c). So evaluating f(2) for f(x) = x³ - 6x² + 11x - 6 gives f(2) = 8 - 24 + 22 - 6 = 0. Since the remainder is 0, (x - 2) is a factor.

Question 4

How do I handle missing terms in the polynomial?

Accepted Answer

If a power of x is missing from the polynomial, use 0 as its coefficient. For example, x³ + 5x - 2 is missing the x² term, so its coefficients are 1, 0, 5, -2. Forgetting to include the zero is the most common mistake in synthetic division.

Question 5

What does it mean when the remainder is zero?

Accepted Answer

A remainder of zero means (x - c) divides the polynomial evenly, so (x - c) is a factor. The quotient then gives the other factor. For example, if x³ - 6x² + 11x - 6 divided by (x - 1) gives quotient x² - 5x + 6 with remainder 0, then the polynomial factors as (x - 1)(x² - 5x + 6) = (x - 1)(x - 2)(x - 3).

Synthetic Division Calculator

How Synthetic Division Works

Worked Example

Frequently Asked Questions