Zeros of Polynomial Functions: College Algebra Study Notes

Polynomial and Rational Functions

Finding the Zeros of a Polynomial Function

In College Algebra, one important skill is determining the zeros (or roots) of polynomial functions. The zeros of a function are the values of the variable that make the function equal to zero.

• Definition: The zero of a function $f(x)$ is any value $x = a$ such that $f(a) = 0$.

• Polynomial Function: A function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$, where $a_n eq 0$.

• Factoring: To find the zeros, set the function equal to zero and solve for $x$. This often involves factoring the polynomial.

Example Problem:

Given the function:

$f(x) = x^3 + 3x^2 - 2x - 8$

Find the zeros of the function.

• Step 1: Set $f(x) = 0$:

$x^3 + 3x^2 - 2x - 8 = 0$

• Step 2: Factor the polynomial (if possible). Try grouping:

Group terms: $(x^3 + 3x^2) + (-2x - 8)$

Factor each group:

• $x^2(x + 3) - 2(x + 4)$

Notice that the grouped terms do not share a common factor, so try rational root theorem or synthetic division.

• Step 3: Use the Rational Root Theorem to test possible rational roots: $ext{Factors of } -8$ divided by $ext{factors of } 1$ (leading coefficient).

• Possible rational roots: $oxed{ ext{±1, ±2, ±4, ±8}}$

Test $x = 1$:

$1^3 + 3(1)^2 - 2(1) - 8 = 1 + 3 - 2 - 8 = -6$ (not zero)

Test $x = -1$:

$(-1)^3 + 3(-1)^2 - 2(-1) - 8 = -1 + 3 + 2 - 8 = -4$ (not zero)

Test $x = 2$:

$2^3 + 3(2)^2 - 2(2) - 8 = 8 + 12 - 4 - 8 = 8$ (not zero)

Test $x = -2$:

$(-2)^3 + 3(-2)^2 - 2(-2) - 8 = -8 + 12 + 4 - 8 = 0$

• So, $x = -2$ is a zero.

Divide the polynomial by $(x + 2)$ using synthetic or long division to find the remaining quadratic factor:

$x^3 + 3x^2 - 2x - 8 ext{ divided by } (x + 2)$ gives $x^2 + x - 4$

Set $x^2 + x - 4 = 0$ and solve using the quadratic formula:

$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-4)}}{2(1)}$

$x = \frac{-1 \pm \sqrt{1 + 16}}{2}$

$x = \frac{-1 \pm \sqrt{17}}{2}$

• Zeros: $x = -2$, $x = \frac{-1 + \sqrt{17}}{2}$, $x = \frac{-1 - \sqrt{17}}{2}$

Summary Table: Zeros of $f(x) = x^3 + 3x^2 - 2x - 8$

Zero	Method
$x = -2$	Rational Root Theorem / Direct Substitution
$x = \frac{-1 + \sqrt{17}}{2}$	Quadratic Formula
$x = \frac{-1 - \sqrt{17}}{2}$	Quadratic Formula

Key Points:

• Always set the function equal to zero to find its zeros.

• Use factoring, synthetic division, or the quadratic formula as appropriate.

• Check all possible rational roots using the Rational Root Theorem.

Additional info: The original question asked for the zeros of a cubic polynomial, which is a standard College Algebra topic under "Polynomial and Rational Functions." The step-by-step solution and table were expanded for clarity and completeness.