Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

4. Polynomial Functions

Zeros of Polynomial Functions

College Algebra

4. Polynomial Functions

Zeros of Polynomial Functions

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1:52 minutes

Problem 1

Textbook Question

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x^3+x^2−4x−4

Verified step by step guidance

Identify the polynomial function: \( f(x) = x^3 + x^2 - 4x - 4 \).

Apply the Rational Zero Theorem, which states that any rational zero, \( \frac{p}{q} \), of the polynomial is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.

Determine the factors of the constant term, \(-4\), which are \( \pm 1, \pm 2, \pm 4 \).

Determine the factors of the leading coefficient, \(1\), which are \( \pm 1 \).

List all possible rational zeros by forming fractions \( \frac{p}{q} \) using the factors of the constant term and the leading coefficient: \( \pm 1, \pm 2, \pm 4 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Zero Theorem

The Rational Zero Theorem states that any rational solution (or zero) of a polynomial equation, expressed in the form p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps in identifying potential rational roots of a polynomial function, which can then be tested for actual roots.

Factors of a Polynomial

Factors of a polynomial are the expressions that can be multiplied together to yield the polynomial. For the Rational Zero Theorem, identifying the factors of the constant term and the leading coefficient is crucial, as these factors determine the possible values of p and q in the rational zeros p/q.

Testing for Zeros

Once the possible rational zeros are identified using the Rational Zero Theorem, each candidate must be tested in the polynomial function to determine if it is indeed a zero. This is typically done by substituting the candidate into the polynomial and checking if the result equals zero, confirming that it is a root of the function.