Lesson 2.13
The Factor Theorem
The Remainder Theorem told us . The Factor Theorem takes it one step further: if the remainder is zero, then is a factor — and you can use division to find the other factor.
Introduction
In Lesson 2.9, you learned the Remainder Theorem: dividing by gives remainder . The Factor Theorem is the special case where that remainder equals zero — meaning divides evenly and is a factor.
Past Knowledge
Remainder Theorem and synthetic division from Lessons 2.8-2.9.
Today's Goal
Use the Factor Theorem to verify roots, and use synthetic division to fully factor a polynomial once a root is known.
Future Success
This theorem is the bridge between "guessing" roots and completely factoring any polynomial.
Key Concepts
The Factor Theorem
For a polynomial and a number :
is a factor of
⟺
The double arrow ⟺ means it works both ways. If one is true, the other is automatically true.
The Two-Step Strategy
Step 1: Verify the Root
Plug into . If , then is a factor.
Step 2: Divide to Find the Other Factor
Use synthetic division by to get the quotient — that's your other factor. Continue factoring the quotient if possible.
Worked Examples
Example 1: Verify and Factor
BasicGiven , show that is a factor, then factor completely.
Evaluate
Synthetic Division by 2
| 2 | 1 | −4 | 1 | 6 |
| 2 | −4 | −6 | ||
| 1 | −2 | −3 | 0 |
Quotient:
Factor the Quotient
Example 2: Given a Zero, Factor Completely
IntermediateFactor given that is a zero.
Synthetic Division by 2
| 2 | 2 | 1 | −13 | 6 |
| 4 | 10 | −6 | ||
| 2 | 5 | −3 | 0 |
Factor the Quotient
Example 3: Degree 4 — Two Rounds of Division
AdvancedFactor given that and are zeros.
First Division: Divide by
| 1 | 1 | −5 | 5 | 5 | −6 |
| 1 | −4 | 1 | 6 | ||
| 1 | −4 | 1 | 6 | 0 |
Quotient:
Second Division: Divide quotient by
| −1 | 1 | −4 | 1 | 6 |
| −1 | 5 | −6 | ||
| 1 | −5 | 6 | 0 |
Quotient:
Final Answer
Common Pitfalls
Confusing the Sign on
If the factor is , then , NOT . Remember: .
Not Factoring Completely
After one round of synthetic division, the quotient may still be factorable. Always check if the remaining quadratic (or higher) can be factored further.
Real-Life Applications
Control systems engineers use the Factor Theorem to find the zeros of transfer functions — polynomial fractions that model how a system responds to input. Knowing the zeros tells them exactly where the system output goes to zero, which is critical for designing stable autopilots, thermostats, and other feedback systems.
Practice Quiz
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