Lesson 5.4

The Rational Zero Theorem

Stop guessing blindly. Use the "p over q" strategy to generate a hit-list of possible roots for any polynomial.

Introduction

Prerequisite Connection: You know how to check if a number is a root (Synthetic Division). But if I give you and ask for the zeros, where do you start? -1? 100? 0.5?

Today's Increment: The Rational Zero Theorem narrows down the infinite possibilities to a short list of "candidates." If a polynomial has a rational root, it MUST be in this list.

Why This Matters: Before computers, this was the primary way algebraic equations were solved. It remains the best way to solve exact equations without relying on decimal approximations.

Explanation of Key Concepts

The "p/q" Rule

Equation

Factors of Constant Term (

)

"The caboose"

Factors of Leading Coeff (

)

"The driver"

Possible Rational Zeros =

The "Search & Destroy" Strategy

List Candidates: Write out all p's and q's. Form fractions.
Test Them: Use Synthetic Division to check for Remainder 0.
Depress: Once you get a zero, write the new (smaller) polynomial.
Repeat: Keep going until you hit a Quadratic (degree 2).
Finish: Use Quadratic Formula or Factoring to find the last two roots.

Worked Examples

Level: Basic

Example 1: Making the List

List all possible rational zeros for .

p (Constant = 6)

Factors: 1, 2, 3, 6

q (Leading = 2)

Factors: 1, 2

Now take every p over every q:

Over 1:
Over 2:

Final Simplified List

. (Duplicates removed)

Level: Intermediate

Example 2: Solve the Polynomial

Find all zeros for .

Step 1: List candidates. q=1, p=6. List:

Step 2: Test candidates. Let's try

-4

↓

-4

-6

-2

-3

0 (Match!)

Step 3: Factor the depressed polynomial.

Zeros:

Common Pitfalls

Mixing up p and q:
Remember: is the Constant (end), is the Leading (start). Think "Constants keep their promises (p)". Or "Prince and Queen" (P on top).
Forgetting the :
Factors can be positive or negative. If you forget the negative ones, you miss half the answers!

Real-World Application

Acoustics: Resonant Frequencies

When designing a speaker enclosure, engineers must solve a "Characteristic Equation" to find the resonant frequencies where the box will vibrate.

These equations are often 3rd or 4th degree polynomials. Solving them involves finding the roots—exactly what we are doing here. The valid frequencies must be real numbers, often rational, which correspond to the dimensions of the box.