Use the Rational Zero Theorem to list all possible rational zeros for each given function. $f\left(x\right)=x^4-6x^3+14x^2-14x+5$

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = x^4 - 6x^3 + 14x^2 - 14x + 5\).
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. Find the factors of the constant term (the last term), which is 5. The factors of 5 are \(\pm 1\) and \(\pm 5\).
4. Find the factors of the leading coefficient (the coefficient of \(x^4\)), which is 1. The factors of 1 are \(\pm 1\).
5. List all possible rational zeros by forming all fractions \(\frac{p}{q}\) using the factors found: \(\pm 1\) and \(\pm 5\) (since dividing by 1 does not change the value). So, the possible rational zeros are \(\pm 1\) and \(\pm 5\).

Key Concepts

Rational Zero Theorem

The Rational Zero Theorem provides a list of all possible rational zeros of a polynomial function by considering factors of the constant term and the leading coefficient. Specifically, any rational zero is of the form ±(factor of constant term) / (factor of leading coefficient). This theorem helps narrow down candidates for zeros before testing them.

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. Zeros of a polynomial are values of x that make the function equal to zero. Finding zeros is essential for graphing and solving polynomial equations, and rational zeros are a subset that can be expressed as fractions.

Factoring and Testing Possible Zeros

After listing possible rational zeros using the Rational Zero Theorem, each candidate must be tested by substitution or synthetic division to determine if it is an actual zero. Factoring the polynomial using confirmed zeros simplifies the function and helps find all roots, including irrational or complex ones.