In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x^4−11x^3−3x^2−6x+8

Verified step by step guidance

1. Identify the leading coefficient and the constant term of the polynomial. Here, the leading coefficient is 3 and the constant term is 8.
2. List the factors of the constant term (8). These are: \( \pm 1, \pm 2, \pm 4, \pm 8 \).
3. List the factors of the leading coefficient (3). These are: \( \pm 1, \pm 3 \).
4. Use the Rational Zero Theorem, which states that any rational zero of the polynomial is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
5. List all possible rational zeros by forming fractions \( \frac{p}{q} \) using the factors from steps 2 and 3. Simplify the fractions to get the final list of possible rational zeros.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem states that any rational solution (or zero) of a polynomial equation, expressed in the form p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps in identifying all possible rational zeros of a polynomial function, which can then be tested to find actual zeros.

Factors of a Polynomial

To apply the Rational Zero Theorem, one must first determine the factors of the constant term and the leading coefficient of the polynomial. For the function f(x) = 3x^4 - 11x^3 - 3x^2 - 6x + 8, the constant term is 8 and the leading coefficient is 3. The factors of these numbers are used to generate potential rational zeros.

Testing Possible Zeros

Once the possible rational zeros are identified using the Rational Zero Theorem, each candidate must be tested in the polynomial function to determine if it is indeed a zero. This is typically done by substituting the candidate into the polynomial and checking if the result equals zero. Successful candidates indicate actual zeros of the polynomial.