Lesson 5.3

The Factor Theorem

The bridge between algebra and geometry. Connecting roots, zeros, and factors into one unified concept.

Introduction

The bridge between algebra and geometry. Connecting roots, zeros, and factors into one unified concept.

Past Knowledge

We know that . We also know that if a number divides evenly with no remainder, it is called a "factor" (like how 3 is a factor of 12).

Today's Goal

We combine these two ideas. If , then the remainder is 0, which means divides perfectly into the polynomial.

Future Success

Factoring is easy. Factoring is hard. The Factor Theorem gives us a methodical way to break down high-degree polynomials to find limits and vertical asymptotes.

Key Concepts

The Biconditional Truth

is a factor of
If and Only If
Algebraic Meaning:
If you plug in and get 0, synthetic division will have a 0 remainder.
Geometric Meaning:
If is a factor, the graph crosses the x-axis at .

Worked Examples

Level: Basic

Example 1: Is it a Factor?

Is a factor of ?

Method 1: Plug it in. Calculate .


Result
Yes! Since , is a factor.
Level: Intermediate

Example 2: Depressing the Polynomial

Given that is a root of , factor the polynomial completely.

Since 2 is a root, is a factor. Divide it out to find the rest.
2
1
0 (0x²)
-7
6
↓
2
4
-6
1
2
-3
0
The "Depressed Polynomial" is .
Now factor the quadratic: .
Final Answer:
Level: Advanced

Example 3: Unknown Coefficient

Find so that is a factor of .

Step 1: Set up Condition
factor → .



So .

Common Pitfalls

  • Confusing Roots and Factors:

    Root: . Factor: .
    Root: . Factor: .
    Always flip the sign!

  • Thinking Remainder 0 means x=0:

    No! Remainder 0 means the y-value is 0. It means you FOUND an x-intercept.

Real-Life Applications

Error Correction Codes (Reed-Solomon)

QR codes and CDs use the Factor Theorem to fix scratches and smudges.

They encode data as a polynomial designed to have specific roots (like ). When the scanner reads the code, it calculates . If it gets 0, the data is clean. If it gets a number like 5, the "remainder" tells the scanner exactly where the error is and how to fix it!

Practice Quiz

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