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 - 1. (simplify your answer.) c. use the zero from part (b) to find all the zeros of the polynomial function. the zeros of the function are . (simplify your answer. use a comma to separate answers as needed. type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Recall synthetic - division result

Since - 1 is a zero of \(f(x)=x^{3}+5x^{2}-30x - 34\), we use synthetic division to divide \(x^{3}+5x^{2}-30x - 34\) by \(x + 1\). The synthetic - division setup is:

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

The quotient is \(x^{2}+4x - 34\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 4\), \(c=-34\)), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute the values: \(x=\frac{-4\pm\sqrt{4^{2}-4\times1\times(-34)}}{2\times1}=\frac{-4\pm\sqrt{16 + 136}}{2}=\frac{-4\pm\sqrt{152}}{2}=\frac{-4\pm2\sqrt{38}}{2}=-2\pm\sqrt{38}\).
\(\sqrt{38}\approx6.164\), so \(x=-2+\sqrt{38}\approx - 2+6.164 = 4.164\) and \(x=-2-\sqrt{38}\approx-2 - 6.164=-8.164\).

Answer:

\(-1,4.164,-8.164\)