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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2:02 minutes

Problem 3

Textbook Question

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−x^2+19x+6

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Identify the leading coefficient and the constant term of the polynomial. Here, the leading coefficient is 3 and the constant term is 6.

List all factors of the constant term (6). These are: ±1, ±2, ±3, ±6.

List all factors of the leading coefficient (3). These are: ±1, ±3.

Use the Rational Zero Theorem, which states that any rational zero of the polynomial is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Combine the factors to list all possible rational zeros: ±1, ±2, ±3, ±6, ±1/3, ±2/3.

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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 all possible rational zeros of a polynomial function, which can then be tested to find actual zeros.

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 (the term without a variable) and the leading coefficient (the coefficient of the highest degree term) is essential, as these factors determine the potential rational zeros of the polynomial.

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, which confirms that the candidate is a valid zero of the function.