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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 6
Chapter 4, Problem 6

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−3x2−6x+8

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Identify the polynomial function: f(x)=3x4−11x3−3x2−6x+8.
Recall the Rational Zero Theorem: possible rational zeros are of the form \(\frac{p}{q}\), where p is a factor of the constant term and q is a factor of the leading coefficient.
List the factors of the constant term (8): \(\pm\) 1, \(\pm\) 2, \(\pm\) 4, \(\pm\) 8.
List the factors of the leading coefficient (3): \(\pm\) 1, \(\pm\) 3.
Form all possible rational zeros by taking each factor of 8 over each factor of 3, simplifying if possible, to get the complete list of candidates.

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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 provides a method to list all possible rational zeros of a polynomial function. It states that any rational zero, expressed as a fraction p/q in lowest terms, must have p as a factor of the constant term and q as a factor of the leading coefficient.
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Factors of Integers

To apply the Rational Zero Theorem, you need to find all factors of the constant term and the leading coefficient. Factors are integers that divide the number without leaving a remainder. These factors help generate all possible rational zeros by forming fractions p/q.
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Polynomial Functions and Zeros

A zero of a polynomial function is a value of x that makes the function equal to zero. Understanding zeros is essential for solving polynomial equations and analyzing their graphs. Rational zeros are specific zeros that can be expressed as fractions of integers.
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Related Practice
Textbook Question

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

f(x)=x2+2x+1g(x)=x2−2x+1f(x) = x^2 + 2x + 1 \(\quad\) g(x) = x^2 - 2x + 1

h(x)=x2−1j(x)=−x2−1h(x) = x^2 - 1 \(\quad\) j(x) = -x^2 - 1

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Textbook Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x5−x4−7x3+7x2−12x−12

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x)r(x). (6x3+7x2+12x−5)÷(3x−1)(6x^3+7x^2+12x-5)\(\div\)(3x-1)

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Textbook Question

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x4−x3+5x2−2x−6

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Divide using long division. State the quotient, and the remainder, r(x)r(x). (12x2+x−4)÷(3x−2)(12x^2+x-4)\(\div\)(3x-2)

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Textbook Question

Find the domain of each rational function. h(x)=(x+7)/(x2−49)

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