# Nakafa Framework: LLM

URL: /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 <InlineMath math="P(x)" /> by <InlineMath math="(x - c)" />, sometimes the remainder is zero. We know from the Remainder Theorem that if the remainder is zero, then <InlineMath math="P(c) = 0" />. So, what does it mean if <InlineMath math="P(c) = 0" />?

The value <InlineMath math="c" /> that causes <InlineMath math="P(c) = 0" /> is called a **zero** or a **root** of the polynomial <InlineMath math="P(x)" />. The Factor Theorem explains the close relationship between these zeros and the factors of the polynomial.

### Statement of the Factor Theorem

Let <InlineMath math="P(x)" /> be a polynomial and <InlineMath math="c" /> be a real number.

<InlineMath math="(x - c)" /> is a **factor** of <InlineMath math="P(x)" /> **if and only if** <InlineMath math="P(c) = 0" />.

This is a two-way statement:

1.  **If <InlineMath math="(x - c)" /> is a factor of <InlineMath math="P(x)" />, then <InlineMath math="P(c) = 0" />.**

    (If a number is perfectly divisible by another number, the remainder must be zero).

2.  **If <InlineMath math="P(c) = 0" />, then <InlineMath math="(x - c)" /> is a factor of <InlineMath math="P(x)" />.**

    (If the value of the polynomial at <InlineMath math="x=c" /> is zero, it means <InlineMath math="(x-c)" /> 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:

<BlockMath math="P(x) = (x - c) \cdot H(x) + S" />

And from the Remainder Theorem, we know <InlineMath math="S = P(c)" />.

<BlockMath math="P(x) = (x - c) \cdot H(x) + P(c)" />

- If <InlineMath math="(x - c)" /> is a factor, it means <InlineMath math="P(x)" /> is divisible by <InlineMath math="(x - c)" />. This only happens if the remainder is zero. Thus, <InlineMath math="S = P(c) = 0" />.
- If <InlineMath math="P(c) = 0" />, then <InlineMath math="S = 0" />. The equation becomes <InlineMath math="P(x) = (x - c) \cdot H(x) + 0" />, or <InlineMath math="P(x) = (x - c) \cdot H(x)" />. This shows that <InlineMath math="(x - c)" /> is a factor of <InlineMath math="P(x)" />.

## 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 <InlineMath math="c" /> such that <InlineMath math="P(c) = 0" />.
2.  **Confirm Factor:** If <InlineMath math="P(c) = 0" />, then according to the Factor Theorem, <InlineMath math="(x - c)" /> is a factor of <InlineMath math="P(x)" />.
3.  **Divide the Polynomial:** Use Horner's method or long division to divide <InlineMath math="P(x)" /> by the factor <InlineMath math="(x - c)" /> found. The quotient is <InlineMath math="H(x)" />.

    <BlockMath math="P(x) = (x - c) \cdot H(x)" />

4.  **Factor the Quotient:** If <InlineMath math="H(x)" /> can still be factored (e.g., if <InlineMath math="H(x)" /> is a quadratic or cubic polynomial whose roots can be found), repeat the process starting from step 1 on <InlineMath math="H(x)" />.
5.  **Complete Factorization:** Write <InlineMath math="P(x)" /> as the product of all the linear factors found.

### Factoring a Polynomial

Let <InlineMath math="P(x) = x^3 + 2x^2 - 13x + 10" />. We notice that the sum of all coefficients and the constant (<InlineMath math="1 + 2 - 13 + 10" />) is 0. This indicates that <InlineMath math="P(1) = 0" />.

1.  **Confirm Zero:**

    Calculate <InlineMath math="P(1)" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="P(1) = (1)^3 + 2(1)^2 - 13(1) + 10" />
      <BlockMath math="P(1) = 1 + 2 - 13 + 10" />
      <BlockMath math="P(1) = 0" />
    </div>

2.  **Confirm Factor:**
    
    Since <InlineMath math="P(1) = 0" />, <InlineMath math="(x - 1)" /> is a factor of <InlineMath math="P(x)" />.

3.  **Divide Polynomial:**

    We use Horner's method to divide <InlineMath math="P(x)" /> by <InlineMath math="(x - 1)" /> (<InlineMath math="c = 1" />).

    <BlockMath
      math="
    \begin{array}{c|cccc}
    1 & 1 & 2 & -13 & 10 \\
      &   & 1 & 3 & -10 \\
    \hline
      & 1 & 3 & -10 & \boxed{0} \\
    \end{array}
    "
    />

    The quotient is <InlineMath math="H(x) = 1x^2 + 3x - 10 = x^2 + 3x - 10" />. The remainder is 0, as expected.

    So, <InlineMath math="P(x) = (x - 1)(x^2 + 3x - 10)" />.

4.  **Factor the Quotient:**

    Factor the quadratic polynomial <InlineMath math="H(x) = x^2 + 3x - 10" />.

    <BlockMath math="x^2 + 3x - 10 = (x - 2)(x + 5)" />

5.  **Complete Factorization:**

    Combine all factors.

    <BlockMath math="P(x) = (x - 1)(x - 2)(x + 5)" />

## Exercise

Let <InlineMath math="P(x) = x^3 - 2x^2 - 21x - 18" />. Show that <InlineMath math="P(-1) = 0" />, and use this to factor <InlineMath math="P(x)" /> completely.

### Answer Key

1.  **Show <InlineMath math="P(-1) = 0" />:**

    <div className="flex flex-col gap-4">
      <BlockMath math="P(-1) = (-1)^3 - 2(-1)^2 - 21(-1) - 18" />
      <BlockMath math="P(-1) = -1 - 2(1) + 21 - 18" />
      <BlockMath math="P(-1) = -1 - 2 + 21 - 18" />
      <BlockMath math="P(-1) = -3 + 3 = 0" />
    </div>
    Proven <InlineMath math="P(-1) = 0" />.

2.  **Confirm Factor:**
    
    Since <InlineMath math="P(-1) = 0" />, <InlineMath math="(x - (-1)) = (x + 1)" /> is a factor of <InlineMath math="P(x)" />.

3.  **Divide Polynomial (Horner's Method with <InlineMath math="c = -1" />):**

    <BlockMath
      math="
    \begin{array}{c|cccc}
    -1 & 1 & -2 & -21 & -18 \\
       &   & -1 & 3 & 18 \\
    \hline
       & 1 & -3 & -18 & \boxed{0} \\
    \end{array}
    "
    />

    The quotient is <InlineMath math="H(x) = x^2 - 3x - 18" />.

    So, <InlineMath math="P(x) = (x + 1)(x^2 - 3x - 18)" />.

4.  **Factor the Quotient:**

    Factor <InlineMath math="H(x) = x^2 - 3x - 18" />.

    <BlockMath math="x^2 - 3x - 18 = (x - 6)(x + 3)" />

5.  **Complete Factorization:**

    <BlockMath math="P(x) = (x + 1)(x - 6)(x + 3)" />
