How to use the rational zero theorem?

Understand the Problem

The question is asking for guidance on how to apply the rational zero theorem, which is a concept in mathematics used to find possible rational roots of polynomial equations. We'll provide a brief overview of the theorem and steps to use it effectively.

Answer

The Rational Zero Theorem provides a method to identify possible rational zeros of polynomials as $\frac{p}{q}$, where $p$ are factors of the constant and $q$ are factors of the leading coefficient.

Steps to Solve

State the Rational Zero Theorem

The Rational Zero Theorem states that any rational solution (or zero) of the polynomial equation $$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0$$ must be of the form $\frac{p}{q}$, where:

$p$ is a factor of the constant term $a_0$.
$q$ is a factor of the leading coefficient $a_n$.

Identify the Polynomial

Identify your polynomial equation and determine its leading coefficient $a_n$ and constant term $a_0$. For example, if you have the polynomial $2x^3 - 3x^2 + 4x - 6$, then $a_n = 2$ and $a_0 = -6$.

List the Factors of $a_0$ and $a_n$

Find all the factors of the constant term and leading coefficient:

For $a_0 = -6$: the factors are $\pm 1, \pm 2, \pm 3, \pm 6$.
For $a_n = 2$: the factors are $\pm 1, \pm 2$.

Form Possible Rational Zeros

Create a list of all possible values for $\frac{p}{q}$ using the factors of $a_0$ and $a_n$. For this example:

Possible rational zeros = $\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 2}, \frac{\pm 2}{\pm 1}, \frac{\pm 2}{\pm 2}, \frac{\pm 3}{\pm 1}, \frac{\pm 3}{\pm 2}, \frac{\pm 6}{\pm 1}, \frac{\pm 6}{\pm 2}$.

This results in: $${-6, -3, -2, -1, -\frac{3}{2}, -\frac{1}{2}, 1, 2, 3, 6, \frac{3}{2}, \frac{1}{2}}$$

Test Possible Zeros

Substitute each of these possible rational zeros back into the polynomial to determine if they result in zero. This is done by plugging the values into the polynomial equation until you find the actual roots.

The possible rational zeros of a polynomial can be found using the Rational Zero Theorem.

More Information

Finding rational roots is an important step in polynomial factorization, which can simplify solutions to polynomial equations. The roots help uncover the behavior of polynomials, like identifying intervals where they are positive or negative.

Tips

Forgetting to consider both positive and negative factors of $a_0$ and $a_n$.
Not reducing the fraction $p/q$ to its simplest form.
Failing to test all possible rational roots to find actual roots.

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