Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x⁴−x³+5x²−2x−6

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Identify the polynomial function: \(f(x) = 4x^{4} - x^{3} + 5x^{2} - 2x - 6\).

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.

List the factors of the constant term \(-6\): \(\pm1, \pm2, \pm3, \pm6\).

List the factors of the leading coefficient \(4\): \(\pm1, \pm2, \pm4\).

Form all possible rational zeros by taking each factor of the constant term over each factor of the leading coefficient, simplifying if possible, and include both positive and negative values.

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Rational Zero Theorem

The Rational Zero Theorem helps identify all possible rational zeros of a polynomial function by considering factors of the constant term and the leading coefficient. Possible rational zeros are formed by ±(factors of constant term) divided by (factors of leading coefficient). This theorem narrows down candidates for roots before testing them.

Polynomial Functions and Degree

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The degree of the polynomial is the highest exponent, which determines the maximum number of zeros the function can have. Understanding the degree helps anticipate the number of possible roots.

Factors of Integers

Factors of an integer are numbers that divide it exactly without leaving a remainder. To apply the Rational Zero Theorem, you must find all positive and negative factors of the constant term and leading coefficient. This step is essential to list all possible rational zeros systematically.

Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x4−x3+5x2−2x−6