Lesson 2.9

The Remainder Theorem

A surprisingly elegant shortcut: the remainder when you divide by is exactly . This single idea unlocks efficient root-finding.

Introduction

Instead of plugging into a long polynomial and simplifying, you can use synthetic division and just look at the remainder. This saves time and connects division to evaluation.

Past Knowledge

You know how to divide using synthetic division and find the remainder.

Today's Goal

Use the Remainder Theorem to evaluate and determine if is a factor.

Future Success

The Factor Theorem (a special case) is the key to finding all zeros of a polynomial in the next chapter.

Key Concepts

1. The Remainder Theorem

When a polynomial is divided by , the remainder equals .

where

This means you can evaluate using synthetic division instead of direct substitution!

2. Two Methods, Same Answer

Method A: Direct Substitution

Plug into and simplify.

Method B: Synthetic Division

Divide by and read off the remainder.

3. The Factor Theorem

A special case: if the remainder is , then is a factor and is a zero (root) of the polynomial.

→ IS a factor
≠ → is NOT a factor

💡 Why This is Powerful

Later, the Rational Root Theorem will give us a list of candidate zeros. We'll use the Remainder/Factor Theorem to quickly test each candidate using synthetic division.

Worked Examples

Example 1: Evaluate Using Both Methods

Basic

Find if .

Method A: Substitution

Method B: Synthetic Division

21−304
2−2−4
1−1−20

, so is a factor and is a zero.

Example 2: Is (x − 3) a Factor?

Intermediate

Determine if is a factor of .

1

Strategy

By the Factor Theorem, is a factor if and only if . Use synthetic division with .

2

Synthetic Division

31−214
3312
11416
✓

Conclusion

≠ 0, so NO — is not a factor.

The remainder is 16, meaning is not a zero of this polynomial.

Example 3: Finding an Unknown Coefficient

Advanced

Find the value of so that is a factor of .

1

Apply the Factor Theorem

is a factor means .

2

Substitute

3

Solve for

So and is a factor.

Common Pitfalls

Confusing the Remainder with the Quotient

The Remainder Theorem says equals the remainder, not the quotient. The bottom row of synthetic division gives both — the last number is the remainder, and everything before it is the quotient.

Factor ≠ Zero (Sign Confusion)

If is a factor, then is the zero (not ). The factor and the zero always have opposite signs.

Real-Life Applications

In quality control, a polynomial model might predict the output of a machine at time . The Remainder Theorem lets engineers quickly check: "At , is the output exactly zero?" If so, that time value is a critical point. This is faster than evaluating the entire polynomial by hand, especially for high-degree models.

Practice Quiz

Loading...