Question

the following equation is given. complete parts (a)–(c).
$x^{3}-3x^{2}-25x + 75 = 0$
a. list all rational roots that are possible according to the rational zero theorem.
(use a comma to separate answers as needed.)

Explanation:

Step1: Identify the leading - coefficient and constant term

For the polynomial \(f(x)=x^{3}-3x^{2}-25x + 75\), the leading - coefficient \(a_{n}=1\) and the constant term \(a_{0}=75\).

Step2: Find factors of leading - coefficient and constant term

The factors of the leading - coefficient \(a_{n}=1\) are \(\pm1\). The factors of the constant term \(a_{0}=75\) are \(\pm1,\pm3,\pm5,\pm15,\pm25,\pm75\).

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 \(a_{n}\).
Since \(q = \pm1\) and \(p=\pm1,\pm3,\pm5,\pm15,\pm25,\pm75\), the possible rational roots are \(\pm1,\pm3,\pm5,\pm15,\pm25,\pm75\).

Answer:

\(1, - 1,3, - 3,5, - 5,15, - 15,25, - 25,75, - 75\)