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Chapter 3: Problem 65

Describe how to find the possible rational zeros of a polynomial function.

Short Answer

Expert verified
The possible rational zeros of a polynomial function can be found using the Rational Root Theorem. It involves listing all the factor pairs of the constant and leading coefficient, making fractions, and checking whether these fractions when substituted into the function make it zero.

Step by step solution

01

Understanding Rational Root Theorem

Identify the value of the leading coefficient (the number in front of the highest degree term) and the constant term (the number term without a variable) in the polynomial function. Well, according to the Rational Root Theorem, the possible rational zeros of a polynomial function can be determined by listing all the factor pairs of the constant term (as numerators) and the leading coefficient (as denominators).
02

Finding Factors

To make the pairs, list out all the factors of the constant term and the leading coefficient. Remember, factors include both positive and negative numbers because a positive times a negative gives a negative, which might be needed to reach zero in the polynomial equation.
03

Creating Rational Pairs

Form all the possible fractions from the positive and negative factors of the constant term (in the numerator position) and the leading coefficient (in the denominator position). Each of these fractions represents a potential rational zero of the polynomial function.
04

Verification

For verification, these potential zeros can be substituted back into the original polynomial function to check whether they actually make the function zero. If \(f(x) = 0\), it confirms that x is a rational root of the function.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

Whe lise a graphing utility to graph $$ f(x)=\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)=\frac{x^{2}-5 x+6}{x-2} $$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?

Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Kinetic energy varies jointly as the mass and the square of the velocity. A mass of 8 grams and velocity of 3 centimeters per second has a kinetic energy of 36 ergs. Find the kinetic energy for a mass of 4 grams and velocity of 6 centimeters per second.

Explain why a polynomial function of degree 20 cannot cross the \(x\) -axis exactly once.
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