October 6, 2025

What Is The Remainder Theorem

Q: What is the relationship between the Remainder Theorem and the Factor Theorem?

The Factor Theorem is a direct consequence of the Remainder Theorem. It states that (x - c) is a factor of the polynomial ƒ(x) if and only if the remainder, ƒ(c), is equal to zero.

Q: Can the Remainder Theorem be used for divisors like (2x - 1)?

Yes, but you must first set the divisor to zero to find 'c'. For (2x - 1) = 0, we get x = 1/2. You would then evaluate ƒ(1/2) to find the remainder.

Q: What if the divisor is (x + 4)?

You must express the divisor in the form (x - c). The expression (x + 4) can be written as (x - (-4)). Therefore, you would set c = -4 and evaluate ƒ(-4) to find the remainder.

Q: Is using the Remainder Theorem the same as using synthetic division?

They are closely related methods that yield the same remainder. Synthetic division is an algorithmic process that provides both the quotient and the remainder, while the Remainder Theorem is a direct formula to find only the remainder.

Learn about the Remainder Theorem, a shortcut in algebra for finding the remainder of a polynomial division without performing the full division.

Have More Questions →

What Is the Remainder Theorem?

The Remainder Theorem is a principle in algebra that provides a quick way to find the remainder when a polynomial is divided by a linear expression. It states that if a polynomial, ƒ(x), is divided by a linear factor in the form of (x - c), then the remainder is equal to ƒ(c), the value of the polynomial evaluated at x = c.

Section 2: The Core Principle

The theorem is based on the polynomial division algorithm, which can be expressed as ƒ(x) = (x - c) * q(x) + r, where q(x) is the quotient and r is the remainder. When you substitute 'c' for 'x' in this equation, the term (x - c) becomes (c - c), which is zero. This simplifies the equation to ƒ(c) = 0 * q(c) + r, which leaves ƒ(c) = r. This shows that evaluating the function at 'c' directly gives you the remainder.

Section 3: A Practical Example

Consider the polynomial ƒ(x) = x³ - 2x² + 5x - 8. To find the remainder when dividing by (x - 2), we can use the Remainder Theorem instead of long division. Here, c = 2. We simply calculate ƒ(2): ƒ(2) = (2)³ - 2(2)² + 5(2) - 8 = 8 - 8 + 10 - 8 = 2. Therefore, the remainder is 2.

Section 4: Why Is It Important?

The Remainder Theorem is important because it offers an efficient alternative to the lengthy process of polynomial long division, especially when only the remainder is needed. It is also a foundational concept for the Factor Theorem, which is a critical tool for finding the roots or zeros of a polynomial equation and factoring polynomials.

Frequently Asked Questions

What is the relationship between the Remainder Theorem and the Factor Theorem?

Can the Remainder Theorem be used for divisors like (2x - 1)?

What if the divisor is (x + 4)?

Is using the Remainder Theorem the same as using synthetic division?