Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

To find possible rational zeros of a polynomial function, one can use the Rational Root Theorem, which suggests that all possible rational zeros are fractions, wherein the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient. Both positive and negative variants of these fractions are possible rational zeros.

Step 1: Understanding the Rational Root Theorem

The Rational Root Theorem states that if \( p/q \) is a rational zero of a polynomial \( P(x) \), then \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \). The 'factor' here means a number that is multiplied with another number to obtain the concerned value, like the constant term or leading coefficient.

Step 2: Identify Coefficients

A polynomial looks something like this: \( P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 \). Identify the leading coefficient \( a_n \) and the constant term \( a_0 \).

Step 3: List Factors

Write down all factors of the constant term \( a_0 \) and the leading coefficient \( a_n \) separately. Factors are such values that when multiplied together, give the original value for \( a_n \) and \( a_0 \).

Step 4: Write Down Rational Roots

Form all possible fractions where the numerator is from the list of factors of \( a_0 \) and the denominator is from the list of factors of \( a_n \). These fractions, along with their negative variants, are the possible rational zeros.