Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x⁵−x⁴−7x³+7x²−12x−12

Verified step by step guidance

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 \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.
3. Determine the constant term and its factors: the constant term is \(-12\), so the factors of \(p\) are \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm12\).
4. Determine the leading coefficient and its factors: the leading coefficient is \(1\), so the factors of \(q\) are \(\pm1\).
5. List all possible rational zeros by forming all fractions \(\frac{p}{q}\) using the factors found, which simplifies to the factors of the constant term since \(q=\pm1\).

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 without leaving a remainder, and identifying these helps generate all 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, especially the leading and constant terms, are key to applying the Rational Zero Theorem correctly.