Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2.

(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.

(b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.

I should try x = ±1, ±2, ±2/3, and ±1/3

f(1) = 3-2-7-2 ≠ 0
f(-1) = -3-2+7-2 = 0 , so x+1 is a factor
f(-2) = -24 -8 + 14 - 2 ≠0
f(2) = 24 - 8 - 14 - 2 = 0 , so x-2 is a factor
f(2/3) = 8/9 - 8/9 - 14/3 - 2 ≠ 0
f(-2/3) = -8/9 - 8/9 + 14/3 - 2 ≠0
f(1/3) = 1/9 - 2/9 - 7/9 - 2 ≠ 0
f(-1/3) = -1/9 - 2/9 + 7/3 - 2 = 0 , so (3x+1) is a factor

3x^3 – 2x^2 – 7x – 2 = (3x+1)(x-2)(x+1)

x = -1, -1/3, and 2

(a) To find the possible rational zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2, we can use the Rational Zero Theorem. According to this theorem, if a polynomial has a rational zero p/q, where p is a factor of the constant term (-2) and q is a factor of the leading coefficient (3), then p/q is a possible rational zero of the polynomial.

In this case, the factors of the constant term (-2) are ±1 and ±2, and the factors of the leading coefficient (3) are ±1 and ±3. Therefore, the possible rational zeros of f(x) are:

±1/1, ±1/3, ±2/1, and ±2/3

So, the list of possible rational zeros is {-2, -2/3, -1, -1/3, 1/3, 1, 2/3, 2}.

(b) To find the zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2, we need to find the values of x that make the polynomial equal to zero.

We can start by trying each of the possible rational zeros we obtained in part (a) and see if any of them satisfy f(x) = 0.

Let's start with x = -2:
f(-2) = 3(-2)^3 – 2(-2)^2 – 7(-2) – 2
= -24 + 8 + 14 - 2
= -4

Since f(-2) is not equal to zero, -2 is not a zero of the polynomial.

Next, we try x = -2/3:
f(-2/3) = 3(-2/3)^3 – 2(-2/3)^2 – 7(-2/3) – 2
= -8/3 - 4/9 + 14/3 - 2
= 0

Since f(-2/3) is equal to zero, -2/3 is a zero of the polynomial.

We continue this process for all the other possible rational zeros until we find all the zeros of the polynomial.

By doing the calculations, we find that the zeros of the polynomial f(x) = 3x^3 – 2x^2 – 7x – 2 are:

x = -2/3