# Nakafa Framework: LLM
URL: https://nakafa.com/en/subject/high-school/11/mathematics/polynomial/rational-zero
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/polynomial/rational-zero/en.mdx
Output docs content for large language models.
---
export const metadata = {
title: "Rational Zero Theorem",
description: "Master the Rational Zero Theorem to find polynomial roots efficiently. Learn step-by-step methods with examples to factor high-degree polynomials easily.",
authors: [{ name: "Nabil Akbarazzima Fatih" }],
date: "05/04/2025",
subject: "Polynomial",
};
## Finding Rational Roots of Polynomials
After learning about the [Factor Theorem](/subject/high-school/11/mathematics/polynomial/factor-theorem), we know that finding a factor is the same as finding a zero (root) of the polynomial . But how do we find the value of , especially if the polynomial has a high degree?
Trying out all numbers is certainly not efficient. This is where the **Rational Zero Theorem** (or Rational Root Theorem) comes into play. This theorem helps us narrow down the list of possible rational roots of a polynomial.
## Rational Zero Theorem
Let be a polynomial where all coefficients () are **integers**, with and .
If the polynomial has a rational zero (root) of the form (where and are integers, , and is a fraction in simplest form), then:
- must be a factor of the constant term .
- must be a factor of the leading coefficient .
This theorem only provides a list of **possible** rational roots. Not all values of from the list are necessarily actual roots of the polynomial. We still need to test them.
## Steps for Using the Rational Zero Theorem
Here are the steps to find rational roots using this theorem, often combined with the Factor Theorem:
1. **Identify Coefficients:** Ensure all coefficients () are integers. Identify the constant term and the leading coefficient .
2. **List Factors of :** List all integer factors (positive and negative) of the constant term .
3. **List Factors of :** List all integer factors (positive and negative) of the leading coefficient .
4. **List Possible Roots :** List all possible values of by dividing each factor by each factor . Simplify the fractions and remove duplicates.
5. **Test Possible Roots:** Test each value from the list by substituting it into (using the Remainder Theorem) or using Horner's method. If the result is , then is a rational root, and (or the form ) is a factor (Factor Theorem).
6. **Factor Further:** After finding one rational root , use the quotient from Horner's method to find the remaining roots from the lower-degree polynomial.
### Using the Factor Theorem and Rational Zero Theorem
Factor the polynomial completely.
1. **Identify Coefficients:**
The coefficients are integers. and .
2. **Factors of (from ):**
3. **Factors of (from ):**
4. **Possible Roots :**
Dividing all by yields:
5. **Test Possible Roots:** Let's test some values from the list.
- Try :
- Try :
- Try :
- Try :
Success! So, is a root, and is a factor.
- Alternatively, try :
Success! So, is a root, and is a factor.
6. **Factor Further (using the root ):**
Divide by using Horner's ().
The quotient is .
Thus, .
7. **Factor the Quotient:**
Factor .
8. **Complete Factorization:**
## Exercise
Factor completely using the Rational Zero Theorem and the Factor Theorem.
### Answer Key
1. **Identify Coefficients:** , .
2. **Factors of (from 20):** .
3. **Factors of (from 2):** .
4. **Possible Roots :** .
5. **Test Roots:**
Try .
Since , is a root and is a factor.
6. **Divide using Horner ():**
The quotient is .
.
7. **Factor the Quotient:**
Factor .
8. **Complete Factorization:**