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. (use a comma to separate answers as needed.)

Explanation:

Step1: Identify the constant and leading - coefficient

For the polynomial function \(f(x)=x^{3}-5x^{2}-4x + 20\), the constant term \(p = 20\) and the leading - coefficient \(q=1\).

Step2: Find the factors of \(p\) and \(q\)

The factors of \(p = 20\) are \(\pm1,\pm2,\pm4,\pm5,\pm10,\pm20\), and the factors of \(q = 1\) are \(\pm1\).

Step3: Apply the Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial \(f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}\) has integer coefficients, then the possible rational zeros are of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_{0}\) and \(q\) is a factor of the leading - coefficient \(a_{n}\). Here, \(\frac{p}{q}=\pm1,\pm2,\pm4,\pm5,\pm10,\pm20\).

Answer:

\(\pm1,\pm2,\pm4,\pm5,\pm10,\pm20\)