# 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 <InlineMath math="(x-c)" /> is the same as finding a zero (root) <InlineMath math="c" /> of the polynomial <InlineMath math="P(x)" />. But how do we find the value of <InlineMath math="c" />, 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 <InlineMath math="P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0" /> be a polynomial where all coefficients (<InlineMath math="a_n, a_{n-1}, \dots, a_1, a_0" />) are **integers**, with <InlineMath math="a_n \neq 0" /> and <InlineMath math="a_0 \neq 0" />.

If the polynomial <InlineMath math="P(x)" /> has a rational zero (root) of the form <InlineMath math="\frac{p}{q}" /> (where <InlineMath math="p" /> and <InlineMath math="q" /> are integers, <InlineMath math="q \neq 0" />, and <InlineMath math="\frac{p}{q}" /> is a fraction in simplest form), then:

- <InlineMath math="p" /> must be a factor of the constant term <InlineMath math="a_0" />.
- <InlineMath math="q" /> must be a factor of the leading coefficient <InlineMath math="a_n" />.

This theorem only provides a list of **possible** rational roots. Not all values of <InlineMath math="\frac{p}{q}" /> 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 (<InlineMath math="a_n, \dots, a_0" />) are integers. Identify the constant term <InlineMath math="a_0" /> and the leading coefficient <InlineMath math="a_n" />.
2.  **List Factors of <InlineMath math="p" />:** List all integer factors (positive and negative) of the constant term <InlineMath math="a_0" />.
3.  **List Factors of <InlineMath math="q" />:** List all integer factors (positive and negative) of the leading coefficient <InlineMath math="a_n" />.
4.  **List Possible Roots <InlineMath math="\frac{p}{q}" />:** List all possible values of <InlineMath math="\frac{p}{q}" /> by dividing each factor <InlineMath math="p" /> by each factor <InlineMath math="q" />. Simplify the fractions and remove duplicates.
5.  **Test Possible Roots:** Test each value <InlineMath math="\frac{p}{q}" /> from the list by substituting it into <InlineMath math="P(x)" /> (using the Remainder Theorem) or using Horner's method. If the result is <InlineMath math="P(\frac{p}{q}) = 0" />, then <InlineMath math="\frac{p}{q}" /> is a rational root, and <InlineMath math="(x - \frac{p}{q})" /> (or the form <InlineMath math="(qx - p)" />) is a factor (Factor Theorem).
6.  **Factor Further:** After finding one rational root <InlineMath math="c" />, 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 <InlineMath math="P(x) = x^3 + 2x^2 - 9x - 18" /> completely.

1.  **Identify Coefficients:**
    
    The coefficients are integers. <InlineMath math="a_0 = -18" /> and <InlineMath math="a_n = 1" />.

2.  **Factors of <InlineMath math="p" /> (from <InlineMath math="a_0 = -18" />):**

    <BlockMath math="\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18" />

3.  **Factors of <InlineMath math="q" /> (from <InlineMath math="a_n = 1" />):**

    <BlockMath math="\pm 1" />

4.  **Possible Roots <InlineMath math="\frac{p}{q}" />:**

    Dividing all <InlineMath math="p" /> by <InlineMath math="q = \pm 1" /> yields:

    <BlockMath math="\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18" />

5.  **Test Possible Roots:** Let's test some values from the list.

    - Try <InlineMath math="x = 1" />:

      <BlockMath math="P(1) = 1 + 2 - 9 - 18 = -24 \neq 0" />

    - Try <InlineMath math="x = -1" />:

      <BlockMath math="P(-1) = -1 + 2 + 9 - 18 = -8 \neq 0" />

    - Try <InlineMath math="x = 2" />:

      <BlockMath math="P(2) = 8 + 8 - 18 - 18 = -20 \neq 0" />

    - Try <InlineMath math="x = -2" />:

      <BlockMath math="P(-2) = -8 + 8 + 18 - 18 = 0" />
      
      Success! So, <InlineMath math="x = -2" /> is a root, and <InlineMath math="(x+2)" /> is a factor.

    - Alternatively, try <InlineMath math="x = 3" />:

      <BlockMath math="P(3) = (3)^3 + 2(3)^2 - 9(3) - 18 = 27 + 18 - 27 - 18 = 0" />
      
      Success! So, <InlineMath math="x = 3" /> is a root, and <InlineMath math="(x-3)" /> is a factor.

6.  **Factor Further (using the root <InlineMath math="x = 3" />):**

    Divide <InlineMath math="P(x)" /> by <InlineMath math="(x-3)" /> using Horner's (<InlineMath math="c = 3" />).

    <BlockMath
      math="
    \begin{array}{c|cccc}
    3 & 1 & 2 & -9 & -18 \\
      &   & 3 & 15 & 18 \\
    \hline
      & 1 & 5 & 6 & \boxed{0} \\
    \end{array}
    "
    />

    The quotient is <InlineMath math="H(x) = x^2 + 5x + 6" />.
    
    Thus, <InlineMath math="P(x) = (x-3)(x^2 + 5x + 6)" />.

7.  **Factor the Quotient:**

    Factor <InlineMath math="x^2 + 5x + 6" />.

    <BlockMath math="x^2 + 5x + 6 = (x+2)(x+3)" />

8.  **Complete Factorization:**

    <BlockMath math="P(x) = (x-3)(x+2)(x+3)" />

## Exercise

Factor <InlineMath math="P(x) = 2x^3 - 3x^2 - 12x + 20" /> completely using the Rational Zero Theorem and the Factor Theorem.

### Answer Key

1.  **Identify Coefficients:** <InlineMath math="a_0 = 20" />, <InlineMath math="a_n = 2" />.
2.  **Factors of <InlineMath math="p" /> (from 20):** <InlineMath math="\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20" />.
3.  **Factors of <InlineMath math="q" /> (from 2):** <InlineMath math="\pm 1, \pm 2" />.
4.  **Possible Roots <InlineMath math="\frac{p}{q}" />:** <InlineMath math="\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm 1/2, \pm 5/2" />.
5.  **Test Roots:**
    
    Try <InlineMath math="x = 2" />.

    <div className="flex flex-col gap-4">
      <BlockMath math="P(2) = 2(2)^3 - 3(2)^2 - 12(2) + 20" />
      <BlockMath math="P(2) = 2(8) - 3(4) - 24 + 20" />
      <BlockMath math="P(2) = 16 - 12 - 24 + 20" />
      <BlockMath math="P(2) = 4 - 4 = 0" />
    </div>

    Since <InlineMath math="P(2)=0" />, <InlineMath math="x=2" /> is a root and <InlineMath math="(x-2)" /> is a factor.

6.  **Divide using Horner (<InlineMath math="c = 2" />):**

    <BlockMath
      math="
    \begin{array}{c|cccc}
    2 & 2 & -3 & -12 & 20 \\
      &   & 4 & 2 & -20 \\
    \hline
      & 2 & 1 & -10 & \boxed{0} \\
    \end{array}
    "
    />

    The quotient is <InlineMath math="H(x) = 2x^2 + x - 10" />.

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

7.  **Factor the Quotient:**

    Factor <InlineMath math="2x^2 + x - 10" />.

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

8.  **Complete Factorization:**

    <div className="flex flex-col gap-4">
      <BlockMath math="P(x) = (x-2)(2x+5)(x-2)" />
      <BlockMath math="P(x) = (x-2)^2 (2x+5)" />
    </div>
