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 is any value such that .

• Polynomial Function: A function of the form , where .

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

Example Problem:

Given the function:

Find the zeros of the function.

• Step 1: Set :

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

Group terms:

Factor each group:

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: divided by (leading coefficient).

• Possible rational roots:

Test :

(not zero)

Test :

(not zero)

Test :

(not zero)

Test :

• So, is a zero.

Divide the polynomial by using synthetic or long division to find the remaining quadratic factor:

gives

Set and solve using the quadratic formula:

• Zeros: , ,

Summary Table: Zeros of

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.