Question

use the following function to answer parts a through c. f(x)=x^3 + 5x^2 - 30x - 34 a. list all rational zeros that are possible according to the rational zero theorem. 1, - 1,2, - 2,17, - 17,34, - 34 (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) 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 Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial function \(f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}\) 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 \(a_{n}\). For the function \(f(x)=x^{3}+5x^{2}-30x - 34\), \(a_{n}=1\) and \(a_{0}=-34\). The factors of \(a_{0}=-34\) are \(\pm1,\pm2,\pm17,\pm34\) and the factors of \(a_{n} = 1\) is \(\pm1\). So the possible rational zeros are \(\pm1,\pm2,\pm17,\pm34\).

Step2: Use synthetic division to test possible zeros

Let's test \(x = - 1\):

-1 |  1   5   -30   -34
    |     -1   -4    34
    |________________
      1   4   -34     0

Since the remainder is \(0\), \(x=-1\) is a zero of the function \(f(x)\).

Answer:

-1