Find all the zeros of the polynomial function by using the Rational Zero Theorem, Descartes Rule of Signs and Synthetic Division.

f(x)=x^3-3x^2-33x+35

DRS says there are

2 or 0 positive roots
1 negative root

RZT says look for roots among ±1,5,7,35

Synthetic division using easy values first says

f(x) = (x-1)(x^2-2x-35)

I think you can probably take it from there, ok?

To find all the zeros of the polynomial function f(x) = x^3 - 3x^2 - 33x + 35, we will use the Rational Zero Theorem, Descartes' Rule of Signs, and Synthetic Division.

1. Rational Zero Theorem:
The Rational Zero Theorem states that if a polynomial has a rational zero, then it must be of the form p/q, where p is a factor of the constant term (35 in this case) and q is a factor of the leading coefficient (1 in this case).

To find potential rational roots, we need to determine the factors of 35 and 1:
Factors of 35: ±1, ±5, ±7, ±35
Factors of 1: ±1

Therefore, the possible rational roots are: ±1, ±5, ±7, ±35. These are the potential values of x that we will test to check for zeros.

2. Descartes' Rule of Signs:
Descartes' Rule of Signs helps us determine the possible number of positive and negative roots of a polynomial by examining the signs of its coefficients.

Counting the number of sign changes in the equation f(x) = x^3 - 3x^2 - 33x + 35, we have:
- The sign of the constant term is positive (+).
- The sign of the coefficient of the x^3 term is positive (+).
- The sign of the coefficient of the x^2 term is negative (-).
- The sign of the coefficient of the x term is negative (-).

Therefore, by counting the sign changes, we can conclude that there are either 1 positive root or 3 positive roots.

Now, we need to check the sign changes in the equation f(-x):
- The sign of the constant term remains positive (+).
- The sign of the coefficient of the x^3 term remains positive (+).
- The sign of the coefficient of the x^2 term remains negative (-).
- The sign of the coefficient of the x term remains negative (-).

Therefore, by counting the sign changes, we can conclude that there are either 0 negative roots or 2 negative roots.

Based on Descartes' Rule of Signs, we know that there is at least 1 positive root and either 0 or 2 negative roots.

3. Synthetic Division:
Now, we will use synthetic division to test the potential rational roots obtained from the Rational Zero Theorem.
We will start testing with x = 1:
Performing synthetic division with x = 1, we have:

1 | 1 -3 -33 35
| 1 -2 -35
|--------------
| 1 -2 -35 0

Since the remainder is 0, this means x = 1 is a zero or root of the polynomial.

Continuing with synthetic division, we will divide the obtained quotient (1 - 2x - 35x) by x = -1:

-1 | 1 -2 -35
| -1 3
|--------------
| 1 -3 -32

Again, the remainder is 0, so x = -1 is also a zero or root of the polynomial.

By using Descartes' Rule of Signs and checking the remaining potential rational roots ±5, ±7, and ±35, we can conclude that these values are not zeros or roots of the polynomial.

Therefore, the zeros of the polynomial function f(x) = x^3 - 3x^2 - 33x + 35 are x = 1 and x = -1.