Rational Zeros of Polynomials: Practice Test Review

Polynomials and Rational Zeros

Rational Zero Theorem

The Rational Zero Theorem is a fundamental concept in College Algebra that helps identify all possible rational zeros of a polynomial function with integer coefficients. This theorem is especially useful when factoring polynomials or solving polynomial equations.

• Definition: The Rational Zero Theorem states that any rational zero of a polynomial function $P(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$ 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 $a_n$

• Application: To list all possible rational zeros, find all combinations of $\pm p/q$ using the factors of $a_0$ and $a_n$.

Example: Finding Rational Zeros

Given the polynomial:

$P(x) = 2x^3 + 9x^2 + 6x - 8$

• Step 1: Identify the constant term ($a_0 = -8$) and the leading coefficient ($a_n = 2$).

• Step 2: List all factors of $-8$: $\pm1, \pm2, \pm4, \pm8$.

• Step 3: List all factors of $2$:$\pm1, \pm2$.

• Step 4: Form all possible rational zeros $\pm p/q$:

Possible Values for $p$	Possible Values for $q$	Possible Rational Zeros
$\pm1, \pm2, \pm4, \pm8$	$\pm1$	$\pm1, \pm2, \pm4, \pm8$
$\pm1, \pm2, \pm4, \pm8$	$\pm2$	$\pm\frac{1}{2}, \pm1, \pm2, \pm4$ (some overlap)

Final List: $\pm1, \pm\frac{1}{2}, \pm2, \pm4, \pm8$

Key Points

• All possible rational zeros are found by considering every combination of $\pm$ (factor of constant) divided by $\pm$ (factor of leading coefficient).

• Not all listed values are actual zeros; they are candidates to test in the polynomial.

• Use synthetic division or direct substitution to check which candidates are actual zeros.

Example Application

For $P(x) = 2x^3 + 9x^2 + 6x - 8$, the possible rational zeros are:

• $\pm1$

• $\pm\frac{1}{2}$

• $\pm2$

• $\pm4$

• $\pm8$

These values are derived from the factors of $-8$ and $2$ as described above.

Additional info:

• When answering multiple-choice questions, ensure all possible combinations are included.

• Some options may omit fractional candidates; always check for completeness.