Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list.)

P(x) = x3 + 6x2 − 15x − 100

Also, write the polynomial and factor form.

P(x) = (x-4)(x+5)^2

now it's easy, right?

34,8,8

To find the rational zeros of a polynomial, we can use the Rational Root Theorem. According to the theorem, any rational zero of a polynomial with integer coefficients will have the form "p/q", where p is a factor of the constant term and q is a factor of the leading coefficient.

In this case, the leading coefficient is 1 and the constant term is -100. So, the possible rational zeros are integers that can be obtained by dividing any factor of -100 by any factor of 1.

The factors of -100 include: ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.
The factors of 1 include: ±1.

Therefore, the list of possible rational zeros is: ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.

To determine which of these are actual zeros of the polynomial, we can use synthetic division or long division to test each candidate value. Start by dividing the polynomial by (x - p), where p is the candidate value.

For example, let's test if x = 2 is a zero:

Dividing P(x) by (x - 2):
-----------------------------------------
2 | 1 6 -15 -100
- 2 16
---------------
1 8 1 -84
-----------------------------------------

The remainder is -84, so x = 2 is not a zero. Continuing this process for all the possible rational zeros, we find the following zeros:

x = -4, x = 5, and x = -5

Therefore, the rational zeros of the polynomial P(x) = x^3 + 6x^2 - 15x - 100 are -4, 5, and -5.