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Learn the Rational Zero Theorem to find polynomial roots and use candidates to factor high-degree polynomials.
---
## Finding Rational Roots of Polynomials
After learning about the [Factor Theorem](/en/subjects/mathematics/polynomial/factor-theorem), we know that finding a factor $$(x-c)$$ is the same as finding a zero (root) $$c$$ of the polynomial $$P(x)$$. But how do we find the value of $$c$$, especially if the polynomial has a high degree?
Visible text: After learning about the [Factor Theorem](/en/subjects/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 $$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$ be a polynomial where all coefficients ($$a_n, a_{n-1}, \dots, a_1, a_0$$) are **integers**, with $$a_n \neq 0$$ and $$a_0 \neq 0$$.
Visible text: Let be a polynomial where all coefficients () are **integers**, with and .
If the polynomial $$P(x)$$ has a rational zero (root) of the form $$\frac{p}{q}$$ (where $$p$$ and $$q$$ are integers, $$q \neq 0$$, and $$\frac{p}{q}$$ is a fraction in simplest form), then:
Visible text: If the polynomial has a rational zero (root) of the form (where and are integers, , and is a fraction in simplest form), then:
- $$p$$ must be a factor of the constant term $$a_0$$.
- $$q$$ must be a factor of the leading coefficient $$a_n$$.
Visible text: - 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 $$\frac{p}{q}$$ from the list are necessarily actual roots of the polynomial. We still need to test them.
Visible text: 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 ($$a_n, \dots, a_0$$) are integers. Identify the constant term $$a_0$$ and the leading coefficient $$a_n$$.
2. **List Factors of $$p$$:** List all integer factors (positive and negative) of the constant term $$a_0$$.
3. **List Factors of $$q$$:** List all integer factors (positive and negative) of the leading coefficient $$a_n$$.
4. **List Possible Roots $$\frac{p}{q}$$:** List all possible values of $$\frac{p}{q}$$ by dividing each factor $$p$$ by each factor $$q$$. Simplify the fractions and remove duplicates.
5. **Test Possible Roots:** Test each value $$\frac{p}{q}$$ from the list by substituting it into $$P(x)$$ (using the Remainder Theorem) or using Horner's method. If the result is $$P(\frac{p}{q}) = 0$$, then $$\frac{p}{q}$$ is a rational root, and $$(x - \frac{p}{q})$$ (or the form $$(qx - p)$$) is a factor (Factor Theorem).
6. **Factor Further:** After finding one rational root $$c$$, use the quotient from Horner's method to find the remaining roots from the lower-degree polynomial.
Visible text: 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 $$P(x) = x^3 + 2x^2 - 9x - 18$$ completely.
Visible text: Factor the polynomial completely.
1. **Identify Coefficients:**
The coefficients are integers. $$a_0 = -18$$ and $$a_n = 1$$.
2. **Factors of $$p$$ (from $$a_0 = -18$$):**
```math
\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18
```
3. **Factors of $$q$$ (from $$a_n = 1$$):**
```math
\pm 1
```
4. **Possible Roots $$\frac{p}{q}$$:**
Dividing all $$p$$ by $$q = \pm 1$$ yields:
```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 $$x = 1$$:
```math
P(1) = 1 + 2 - 9 - 18 = -24 \neq 0
```
- Try $$x = -1$$:
```math
P(-1) = -1 + 2 + 9 - 18 = -8 \neq 0
```
- Try $$x = 2$$:
```math
P(2) = 8 + 8 - 18 - 18 = -20 \neq 0
```
- Try $$x = -2$$:
```math
P(-2) = -8 + 8 + 18 - 18 = 0
```
Success! So, $$x = -2$$ is a root, and $$(x+2)$$ is a factor.
- Alternatively, try $$x = 3$$:
```math
P(3) = (3)^3 + 2(3)^2 - 9(3) - 18 = 27 + 18 - 27 - 18 = 0
```
Success! So, $$x = 3$$ is a root, and $$(x-3)$$ is a factor.
6. **Factor Further (using the root $$x = 3$$):**
Divide $$P(x)$$ by $$(x-3)$$ using Horner's ($$c = 3$$).
```math
\begin{array}{c|cccc}
3 & 1 & 2 & -9 & -18 \\
& & 3 & 15 & 18 \\
\hline
& 1 & 5 & 6 & \boxed{0} \\
\end{array}
```
The quotient is $$H(x) = x^2 + 5x + 6$$.
Thus, $$P(x) = (x-3)(x^2 + 5x + 6)$$.
7. **Factor the Quotient:**
Factor $$x^2 + 5x + 6$$.
```math
x^2 + 5x + 6 = (x+2)(x+3)
```
8. **Complete Factorization:**
```math
P(x) = (x-3)(x+2)(x+3)
```
Visible text: 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 $$P(x) = 2x^3 - 3x^2 - 12x + 20$$ completely using the Rational Zero Theorem and the Factor Theorem.
Visible text: Factor completely using the Rational Zero Theorem and the Factor Theorem.
### Answer Key
1. **Identify Coefficients:** $$a_0 = 20$$, $$a_n = 2$$.
2. **Factors of $$p$$ (from $$20$$):** $$\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$$.
3. **Factors of $$q$$ (from $$2$$):** $$\pm 1, \pm 2$$.
4. **Possible Roots $$\frac{p}{q}$$:** $$\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20, \pm 1/2, \pm 5/2$$.
5. **Test Roots:**
Try $$x = 2$$.
```math
P(2) = 2(2)^3 - 3(2)^2 - 12(2) + 20
```
```math
P(2) = 2(8) - 3(4) - 24 + 20
```
```math
P(2) = 16 - 12 - 24 + 20
```
```math
P(2) = 4 - 4 = 0
```
Since $$P(2)=0$$, $$x=2$$ is a root and $$(x-2)$$ is a factor.
6. **Divide using Horner ($$c = 2$$):**
```math
\begin{array}{c|cccc}
2 & 2 & -3 & -12 & 20 \\
& & 4 & 2 & -20 \\
\hline
& 2 & 1 & -10 & \boxed{0} \\
\end{array}
```
The quotient is $$H(x) = 2x^2 + x - 10$$.
$$P(x) = (x-2)(2x^2 + x - 10)$$.
7. **Factor the Quotient:**
Factor $$2x^2 + x - 10$$.
```math
2x^2 + x - 10 = (2x+5)(x-2)
```
8. **Complete Factorization:**
```math
P(x) = (x-2)(2x+5)(x-2)
```
```math
P(x) = (x-2)^2 (2x+5)
```
Visible text: 1. **Identify Coefficients:** , .
2. **Factors of (from ):** .
3. **Factors of (from ):** .
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:**