Question

use the following function to answer parts a through c. f(x)=5x^{3}+27x^{2}+11x + 5
a. list all possible rational zeros. 1, - 1,5, - 5,\frac{1}{5},-\frac{1}{5} (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 the possible rational zeros and find an actual zero. one rational zero of the given function is - 5. (simplify your answer.)
c. use the quotient from part (b) to find the remaining zeros of the polynomial function. the remaining zeros of the function are . (simplify your answer. use a comma to separate answers as needed. type an exact answer, using radicals and i as needed.)

Explanation:

Step1: Recall synthetic - division result

Since - 5 is a zero of \(f(x)=5x^{3}+27x^{2}+11x + 5\), when we perform synthetic division with - 5, the coefficients of the polynomial are \(5,27,11,5\). The quotient is a quadratic polynomial.
Performing synthetic - division:

-5 |  5  27  11  5
    |     -25 -10 -5
    |________________
      5   2   1  0

The quotient is \(5x^{2}+2x + 1\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 5\), \(b = 2\), \(c = 1\)), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 5\), \(b = 2\), \(c = 1\) into the formula:
\[

$$\begin{align*} x&=\frac{-2\pm\sqrt{2^{2}-4\times5\times1}}{2\times5}\\ &=\frac{-2\pm\sqrt{4 - 20}}{10}\\ &=\frac{-2\pm\sqrt{- 16}}{10}\\ &=\frac{-2\pm4i}{10}\\ &=\frac{-1\pm2i}{5} \end{align*}$$

\]

Answer:

\(-\frac{1}{5}+\frac{2i}{5},-\frac{1}{5}-\frac{2i}{5}\)