Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x⁴−11x³−x²+19x+6

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = 3x^{4} - 11x^{3} - x^{2} + 19x + 6\).
2. Recall the Rational Zero Theorem: any rational zero, expressed as \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.
3. List the factors of the constant term (6): \(\pm1, \pm2, \pm3, \pm6\).
4. List the factors of the leading coefficient (3): \(\pm1, \pm3\).
5. Form all possible rational zeros by taking each factor of 6 over each factor of 3, simplifying if possible, to get the complete list of candidates.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem provides a method to list all possible rational zeros of a polynomial function. It states that any rational zero, expressed as a fraction p/q in lowest terms, must have p as a factor of the constant term and q as a factor of the leading coefficient.

Factors of Integers

To apply the Rational Zero Theorem, you need to find all factors of the constant term and the leading coefficient. Factors are integers that divide the number exactly without leaving a remainder, and identifying these helps generate all possible rational zeros.

Polynomial Functions and Zeros

A zero of a polynomial function is a value of x that makes the function equal to zero. Understanding how zeros relate to the graph and behavior of polynomials is essential for solving equations and analyzing functions.