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

Polynomial Functions

Polynomial functions are expressions that consist of variables and coefficients, involving operations such as addition, subtraction, and multiplication, as well as non-negative integer exponents of the variables. Understanding the structure of a polynomial is crucial because it influences the behavior of the function, including its roots (zeros), end behavior, and the number of turning points.

Factors of Constant and Leading Coefficient

In the context of the Rational Zero Theorem, identifying the factors of the constant term and the leading coefficient is essential. The factors of the constant term provide possible numerators, while the factors of the leading coefficient provide possible denominators for any candidate rational zero. By considering all possible combinations (with both positive and negative signs), one obtains a list of potential zeros that can be further tested to determine whether they are actual roots of the polynomial.

Rational Zero Theorem

The Rational Zero Theorem is a tool used in algebra to determine a list of all possible rational zeros of a polynomial with integer coefficients. It states that any rational zero, when written in lowest terms as p/q, must have a numerator p that is a factor of the constant term and a denominator q that is a factor of the leading coefficient. This theorem provides a finite set of candidates to test for actual zeros, greatly simplifying the process of finding solutions to polynomial equations.