In Exercises 49–50, use the Rational Zero Theorem to list all possible rational zeros for each g

In Exercises 49–50, use the Rational Zero Theorem to list...

Key Concepts

Factors of Coefficients

In the application of the Rational Zero Theorem, the factors of the constant term and the leading coefficient are essential. By determining all factors of these coefficients, one can generate all possible fractions (p/q) that could serve as rational zeros. This systematic approach allows one to compile a complete list of possible rational roots to test, facilitating the process of factorizing or solving the polynomial.

Rational Zero Theorem

The Rational Zero Theorem is a useful tool for identifying all possible rational zeros of a polynomial with integer coefficients. It states that if a polynomial f(x) has a rational zero expressed in lowest terms as p/q, then p must be a factor of the constant term and q must be a factor of the leading coefficient. This theorem helps streamline the process of finding potential zeros by narrowing the list to easily testable candidates.

Polynomial Function

A polynomial function is an algebraic expression in which the variable is raised to nonnegative integer powers and is multiplied by coefficients. The overall degree of the polynomial (the highest power of the variable) influences its general behavior and the number of potential zeros. Understanding the structure of polynomial functions is crucial when applying techniques like the Rational Zero Theorem.

Transcript

00:01 We need to find the factors of our constant term, as well as the factors of our leading coefficient, so that we can then combine those to get our list of possible rational zeros.

00:10 Constant term would be the term with no variables at all, which in this case would be 15.

00:16 So factors of 15, we know 1 and 15 are factors, because that always works, but also 15 divides by 3 evenly and divides by 5 evenly.

00:27 So our factors would be plus or minus 1, plus or minus 3, plus or minus 5, plus or minus 15.

00:39 All right.

00:40 Now we need to find our factors of our leading coefficient, coefficients being the numbers that come before variables, but specifically a leading coefficient, meaning we need to find the variable with the greatest exponent and use that terms coefficient.

00:55 Well, x to the 4th looks like the highest exponent that i have here, i see an x to the third, x squared, and an x, so x to the first, four would be the greatest, the highest.

01:07 So if x to the fourth is my greatest exponent, then i want that specific coefficient, which in this case happens to be two.

01:16 That means the factors would be one and two, right? i mean, it's two.

01:22 What else is going to be able to divide evenly into that? there's not really going to be much besides those two things, right? so now to find our list of actual possible rational zeros, i'm going to take all the factors of our constant term, and we are going to need to divide each and every one of those by all the factors of our leading coefficient...

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