Questions: Find only the rational zeros. f(x)=x^5-6x^4+8x^3+17x^2-48x+28

Solution Steps

To find the rational zeros of the polynomial \( f(x) = x^5 - 6x^4 + 8x^3 + 17x^2 - 48x + 28 \), we can use the Rational Root Theorem. This theorem suggests that any rational root, expressed as a fraction \(\frac{p}{q}\), has \(p\) as a factor of the constant term (28) and \(q\) as a factor of the leading coefficient (1). We will test these possible rational roots by substituting them into the polynomial and checking if they yield zero.

We are given the polynomial \( f(x) = x^5 - 6x^4 + 8x^3 + 17x^2 - 48x + 28 \).

Using the Rational Root Theorem, we identify the possible rational roots as the factors of the constant term (28) divided by the factors of the leading coefficient (1). The possible rational roots are: \[ \{-28, -14, -7, -4, -2, -1, 1, 2, 4, 7, 14, 28\} \]

We substitute each possible rational root into the polynomial \( f(x) \) to check which values yield \( f(x) = 0 \). The rational roots found are: \[ \{-2, 1, 2\} \]

Final Answer