In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x⁵−x⁴−7x³+7x²−12x−12

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1. Identify the polynomial function: $f(x)=x^5-x^4-7x^3+7x^2-12x-12$.
2. Recall the Rational Zero Theorem: any rational zero, expressed as a fraction $\backslash frac\{p\}\{q\}$, must have $p$ as a factor of the constant term and $q$ as a factor of the leading coefficient.
3. Identify the constant term and the leading coefficient: the constant term is $-12$ and the leading coefficient (coefficient of $x^5$) is $1$.
4. List all factors of the constant term $-12$: $\pm 1,\; \pm 2,\; \pm 3,\; \pm 4,\; \pm 6,\; \pm 12$. Since the leading coefficient is 1, the factors of 1 are $\pm 1$.
5. Form all possible rational zeros by taking each factor of the constant term over each factor of the leading coefficient: this gives the possible rational zeros as $\pm 1,\; \pm 2,\; \pm 3,\; \pm 4,\; \pm 6,\; \pm 12$.

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. This theorem helps narrow down candidates for zeros before testing them.

Factors of Integers

Understanding factors is essential to apply the Rational Zero Theorem. Factors of an integer are numbers that divide it exactly without leaving a remainder. To find possible rational zeros, you must list all factors of the constant term and the leading coefficient, which form the numerator and denominator of potential zeros.

Polynomial Functions and Zeros

A zero of a polynomial function is a value of x that makes the function equal to zero. Identifying zeros helps in factoring and graphing the polynomial. Rational zeros are specific zeros that can be expressed as fractions, and finding them is a key step in solving polynomial equations.