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

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^3+x^2-4x-4$.
2. List the factors of the constant term (the last term) which is -4. The factors are ±1, ±2, ±4.
3. List the factors of the leading coefficient (the coefficient of the highest degree term) which is 1. The factors are ±1.
4. Use the Rational Zero Theorem which states that any possible rational zero is of the form $\backslash frac\{p\}\{q\}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
5. Form all possible rational zeros by dividing each factor of the constant term by each factor of the leading coefficient. Since the leading coefficient is 1, the possible rational zeros are simply ±1, ±2, ±4.

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem provides a list of 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 a Number

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

Polynomial Function Structure

Understanding the structure of a polynomial, including its degree and coefficients, is essential. The degree determines the number of possible zeros, and the coefficients help identify factors needed for the Rational Zero Theorem.