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.
1, - 1,3, - 3,5, - 5,15, - 15,25, - 25,75, - 75
(use a comma to separate answers as needed.)
b. use synthetic division to test several possible rational roots in order to identify one actual root.
one rational root of the given equation is
(simplify your answer.)

Explanation:

Step1: Set up synthetic - division

For the polynomial \(x^{3}-3x^{2}-25x + 75\), the coefficients are \(1,-3,-25,75\). Let's test the possible rational roots from part (a). Let's start with \(x = 3\).

Step2: Perform synthetic - division

Bring down the first coefficient \(1\):

3 |  1  -3  -25  75
   |     3   0 -75
   |----------------
     1   0  -25   0

When we divide \(x^{3}-3x^{2}-25x + 75\) by \(x - 3\) using synthetic - division, the remainder is \(0\).

Answer:

3