f(x)= x^3 - 2x^2 - 21x - 18

all possible rational zeros
# of possible positive zeros
# of possible negative zeros
find all the real zeros

rational zeros (plus or minus)

1,2,3,6,18

By Descartes' Rule of Signs,
1 positive root
2 or 0 negative roots

a little synthetic division yields

f(x) = (x+1)(x+3)(x-6)

To find the possible rational zeros of the polynomial f(x) = x^3 - 2x^2 - 21x - 18, we can use the Rational Root Theorem.

According to the Rational Root Theorem, the possible rational zeros of a polynomial are all the possible ratios of the factors of the constant term (in this case, -18) to the factors of the leading coefficient (in this case, 1).

The factors of -18 are: ±1, ±2, ±3, ±6, ±9, ±18.
The factors of 1 are: ±1.

Therefore, the possible rational zeros are: ±1, ±2, ±3, ±6, ±9, and ±18.

To determine the number of possible positive zeros, we count the number of sign changes in the polynomial f(x) when we substitute positive values for x.

In this case, the polynomial f(x) = x^3 - 2x^2 - 21x - 18 has 2 sign changes when substituting positive values for x. Therefore, there are 2 or 0 positive zeros.

To determine the number of possible negative zeros, we substitute negative values for x and count the number of sign changes.

In this case, the polynomial f(x) = x^3 - 2x^2 - 21x - 18 has 1 sign change when substituting negative values for x. Therefore, there is 1 negative zero.

To find the real zeros, we need to solve the polynomial equation f(x) = 0.

By using numerical methods such as graphing or using a calculator, we can find that the real zeros of the polynomial f(x) = x^3 - 2x^2 - 21x - 18 are approximately x = -1.872, x = 1.245, and x = 9.627.

To find the possible rational zeros of a polynomial function, we can use the Rational Root Theorem. The Rational Root Theorem states that any rational root of a polynomial equation in the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_2x^2 + a_1x + a_0 has the form p/q, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n.

In our case, the polynomial function is f(x) = x^3 - 2x^2 - 21x - 18. The constant term is -18, and the leading coefficient is 1. Therefore, the possible rational zeros are the factors of -18 divided by the factors of 1.

The factors of -18 are: ±1, ±2, ±3, ±6, ±9, ±18
The factors of 1 are: ±1

So, the possible rational zeros are: ±1, ±2, ±3, ±6, ±9, ±18.

To find the number of possible positive zeros, we count the number of positive values from the possible rational zeros. In this case, we count the number of positive values from ±1, ±2, ±3, ±6, ±9, ±18.

There are four positive values: 1, 2, 3, 6. So, there are four possible positive zeros.

To find the number of possible negative zeros, we count the number of negative values from the possible rational zeros. In this case, we count the number of negative values from ±1, ±2, ±3, ±6, ±9, ±18.

There are two negative values: -1, -2. So, there are two possible negative zeros.

To find all the real zeros, we can use various methods, such as factoring, the Rational Root Theorem, or graphing. In this case, we'll use a combination of Rational Root Theorem and synthetic division.

By using synthetic division and checking the possible rational zeros, we find that the real zeros of the given polynomial function f(x) = x^3 - 2x^2 - 21x - 18 are:

x = -3
x = -2
x = 3

Therefore, the real zeros are -3, -2, and 3.