Question

use the rational zero theorem to list all possible rational zeros for the given function. f(x)=5x^4 - 8x^3 + 17x^2 - 17x - 10 choose the answer below that lists all possible rational zeros. oa. -1,1, -5,5, - 1/2,1/2, - 1/10,1/10, - 1/5,1/5, - 5/2,5/2 ob. -1,1, -2,2, -5,5, - 10,10, - 1/5,1/5, - 2/5,2/5 oc. -1,1, -2,2, -5,5, - 10,10, - 1/2,1/2, - 1/5,1/5, - 1/10,1/10 od. -1,1, -2,2, -5,5, - 10,10, - 1/2,1/2, - 1/10,1/10, - 1/5,1/5, - 2/5,2/5, - 5/2,5/2

Explanation:

Step1: Identify leading - coefficient and constant term

The leading - coefficient \(a_n\) of \(f(x)=5x^{4}-8x^{3}+17x^{2}-17x - 10\) is \(a_n = 5\), and the constant term \(a_0=-10\).

Step2: Find factors of leading - coefficient and constant term

The factors of \(a_n = 5\) are \(\pm1,\pm5\), and the factors of \(a_0=-10\) are \(\pm1,\pm2,\pm5,\pm10\).

Step3: Apply the Rational Zero Theorem

The possible rational zeros are of the form \(\frac{p}{q}\), where \(p\) is a factor of \(a_0\) and \(q\) is a factor of \(a_n\).
When \(q = 1\), \(\frac{p}{q}=\pm1,\pm2,\pm5,\pm10\); when \(q = 5\), \(\frac{p}{q}=\pm\frac{1}{5},\pm\frac{2}{5}\).

Answer:

B. \(-1,1,-2,2,-5,5,-10,10,-\frac{1}{5},\frac{1}{5},-\frac{2}{5},\frac{2}{5}\)