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Chapter 2: Problem 65

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

Short Answer

Expert verified
To find the possible rational zeros of a polynomial function, apply the Rational Zero Theorem. This theorem allows one to generate a list of potential rational zeros by taking each factor \( p \) of the constant term over each factor \( q \) of the leading coefficient and considering both positive and negative versions of these fractions. By substituting these potential zeros into the function and checking whether or not they result in an output of zero, the rational zeros of the polynomial can be obtained.

Step by step solution

01

Understand the Rational Zero Theorem

The Rational Zero Theorem provides a list of possible rational zeros of a polynomial function, if there are any. It states that for a polynomial function of \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where all \( a_i \) are integers and \( a_n \neq 0 \), every rational zero of \( P \) will be in the form \( p/q \), where \( p \) is a factor of \( a_0 \) and \( q \) is a factor of \( a_n \).
02

Find the factors of the constant and the leading coefficients

Given a polynomial, identify the constant term (the term with no variable, i.e., \( a_0 \)) and the leading coefficient (the coefficient of the term with the highest degree, i.e., \( a_n \)). List all factors of the constant term and the leading coefficient separately.
03

Generate possible rational zeros

For each pair of integer factor \( p \) of the constant term and \( q \) of the leading coefficient, form the quotient \( p/q \). Consider both \( +p/q \) and \( -p/q \) as potential zeros.
04

Test the potential rational zeros

Plug each potential rational zero into the polynomial to see if the output equals zero. If it does, that means the input is a root of the polynomial. If not, then it is not a root. The list of potential rational zeros that result in zero output are the rational zeros of the given polynomial.

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