In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x⁴−x³+5x²−2x−6

Verified step by step guidance

1. Identify the polynomial function: $f(x)=4x^4\; -\; x^3\; +\; 5x^2\; -\; 2x\; -\; 6$.
2. List the factors of the constant term (the last term), which is -6. The factors are ±1, ±2, ±3, and ±6.
3. List the factors of the leading coefficient (the coefficient of the highest degree term), which is 4. The factors are ±1, ±2, and ±4.
4. Form all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. This gives possible zeros of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides -6 and $q$ divides 4.
5. Write out all unique possible rational zeros: ±1, ±1/2, ±1/4, ±2, ±3, ±3/2, ±3/4, ±6, ±6/2 (which simplifies to ±3), and ±6/4 (which simplifies to ±3/2). Remove duplicates to finalize the list.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem provides a way 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 these factors form the numerator and denominator candidates for 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 helps in solving equations and factoring polynomials, which is essential for analyzing and simplifying polynomial expressions.