Question

the following function is given. f(x)=x^3 - 5x^2 - 4x + 20 a. list all rational zeros that are possible according to the rational zero theorem. ±1,±2,±4,±5,±10,±20 (use a comma to separate answers as needed.) b. use synthetic division to test several possible rational zeros in order to identify one actual zero. one rational zero of the given function is (simplify your answer.)

Explanation:

Step1: Recall synthetic - division setup

For the polynomial \(f(x)=x^{3}-5x^{2}-4x + 20\), and a possible zero \(c\), in synthetic - division, we set up the coefficients \(1,-5,-4,20\) and divide by \(c\).

Step2: Test \(x = 5\)

Set up the synthetic - division with \(c = 5\). Bring down the first coefficient \(1\):
\[

$$\begin{array}{c|cccc} 5&1&-5&-4&20\\ & &5&0&- 20\\ \hline &1&0&-4&0 \end{array}$$

\]
Since the remainder is \(0\), \(x = 5\) is a zero of the polynomial.

Answer:

\(5\)