Polynomial and Rational Functions

Finding Zeros of Polynomial Functions

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. For a polynomial function, these are the solutions to the equation when the function is set equal to zero.

• Definition: The zero of a function is any value such that .

• Polynomial Function: A function of the form , where and is a non-negative integer.

• Factoring: To find the zeros, factor the polynomial and set each factor equal to zero.

Example Problem

Given the function:

Find the zeros of the function.

• Step 1: Set the function equal to zero:

• 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. Test possible rational roots (±1, ±2, ±4, ±8):

• Test :

• Test :

• Test :

• Test :

Therefore, is a zero.

• Step 3: Use synthetic division to factor out :

Divide by :

Result:

• Step 4: Factor using the quadratic formula:

• Final 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 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.