# Nakafa Framework: LLM
URL: https://nakafa.com/en/subject/high-school/11/mathematics/polynomial/factor-theorem
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/polynomial/factor-theorem/en.mdx
Output docs content for large language models.
---
export const metadata = {
title: "Factor Theorem",
description: "Discover how polynomial roots connect to factors. Master the Factor Theorem to find zeros, identify linear factors, and solve complete factorization problems.",
authors: [{ name: "Nabil Akbarazzima Fatih" }],
date: "05/04/2025",
subject: "Polynomial",
};
## Understanding the Factor Theorem
When we divide a polynomial by , sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then . So, what does it mean if ?
The value that causes is called a **zero** or a **root** of the polynomial . The Factor Theorem explains the close relationship between these zeros and the factors of the polynomial.
### Statement of the Factor Theorem
Let be a polynomial and be a real number.
is a **factor** of **if and only if** .
This is a two-way statement:
1. **If is a factor of , then .**
(If a number is perfectly divisible by another number, the remainder must be zero).
2. **If , then is a factor of .**
(If the value of the polynomial at is zero, it means divides the polynomial exactly).
### Connection to the Remainder Theorem
The Factor Theorem is actually a special case of the Remainder Theorem. Recall the division algorithm:
And from the Remainder Theorem, we know .
- If is a factor, it means is divisible by . This only happens if the remainder is zero. Thus, .
- If , then . The equation becomes , or . This shows that is a factor of .
## Using the Factor Theorem to Factor Polynomials
The Factor Theorem is very useful for finding linear factors of a polynomial and then factoring it completely.
**General Steps:**
1. **Find a Zero:** Try guessing or using clues (like the sum of coefficients) to find a value such that .
2. **Confirm Factor:** If , then according to the Factor Theorem, is a factor of .
3. **Divide the Polynomial:** Use Horner's method or long division to divide by the factor found. The quotient is .
4. **Factor the Quotient:** If can still be factored (e.g., if is a quadratic or cubic polynomial whose roots can be found), repeat the process starting from step 1 on .
5. **Complete Factorization:** Write as the product of all the linear factors found.
### Factoring a Polynomial
Let . We notice that the sum of all coefficients and the constant () is 0. This indicates that .
1. **Confirm Zero:**
Calculate .
2. **Confirm Factor:**
Since , is a factor of .
3. **Divide Polynomial:**
We use Horner's method to divide by ().
The quotient is . The remainder is 0, as expected.
So, .
4. **Factor the Quotient:**
Factor the quadratic polynomial .
5. **Complete Factorization:**
Combine all factors.
## Exercise
Let . Show that , and use this to factor completely.
### Answer Key
1. **Show :**
Proven .
2. **Confirm Factor:**
Since , is a factor of .
3. **Divide Polynomial (Horner's Method with ):**
The quotient is .
So, .
4. **Factor the Quotient:**
Factor .
5. **Complete Factorization:**