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The Remainder Theorem

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The Remainder Theorem

Study Guide

Key Definition

The Remainder Theorem states that when a polynomial $f(x)$ is divided by a binomial of the form $x - c$, the remainder is $f(c)$.

Important Notes

Use $f(c)$ to find the remainder when dividing by $x - c$
This theorem simplifies division of polynomials
Evaluate the polynomial at $x = c$ to find the remainder
No need for synthetic or long division
Helps understand polynomial behavior

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$x^2$ is x squared

$\frac{a}{b}$ is a divided by b

$\sqrt{x}$ is the square root of x

Remember to use proper notation when solving problems

Why It Works

The Remainder Theorem provides a quick method to find the remainder of polynomial division by evaluating $f(x)$ at $x = c$.

Remember

For any polynomial $f(x)$, if you divide by $x - c$, the remainder is $f(c)$.

Quick Reference

Remainder Theorem:$f(c)$ is the remainder when $f(x)$ is divided by $x - c$

Understanding The Remainder Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When a polynomial f(x) is divided by x - c, the Remainder Theorem says the remainder is f(c). For example, dividing f(x) = 2x + 3 by x - 1 leaves a remainder f(1) = 5.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the remainder of $f(x) = 5x^3 - 3x^2 + 5x - 3$ when divided by $x - 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard design uses a polynomial function $g(x) = 2x^3 - x^2 + 4x + 5$ for curves. Find the remainder when dividing by $x - 3$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What does the Remainder Theorem tell us about the polynomial $h(x)$ if $h(c) = 0$?

Challenge Quiz

Single Choice Quiz

Advanced

Find the remainder of $p(x) = x^4 - 6x^3 + 11x^2 - 6x$ when divided by $x - 1$.

Recap

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