Determine between which consecutive integers one or more real zeros of f(x)=3x^4+x^3-2x^2+4 are located?

A. no real zeros
B. 0 and 1
C. -2 and -3
D. -1 and 0

I was thinking it's either A or C? I need lots of help.

Graph it on your calculator.

To determine between which consecutive integers one or more real zeros of a polynomial function are located, you can analyze the changes in sign of the function.

In this case, we have the polynomial function f(x) = 3x^4 + x^3 - 2x^2 + 4.

Step 1: Find the critical points
The critical points are the values of x where the function may change its sign. To find the critical points, set the function equal to zero and solve for x:

3x^4 + x^3 - 2x^2 + 4 = 0

Unfortunately, there is no straightforward method to find the exact solutions of a quartic equation (a polynomial of degree 4) algebraically. However, you can use numerical methods or graphing technology to get an approximate idea of the zeros.

Step 2: Use the intermediate value theorem
Alternatively, you can use the Intermediate Value Theorem to identify the intervals where the function changes sign. To do this, evaluate the function at a few points in each interval and check if the signs change.

Let's evaluate the function at x = -3, -2, -1, 0, and 1:

f(-3) = 3(-3)^4 + (-3)^3 - 2(-3)^2 + 4 = 160
f(-2) = 3(-2)^4 + (-2)^3 - 2(-2)^2 + 4 = 4
f(-1) = 3(-1)^4 + (-1)^3 - 2(-1)^2 + 4 = 0
f(0) = 3(0)^4 + (0)^3 - 2(0)^2 + 4 = 4
f(1) = 3(1)^4 + (1)^3 - 2(1)^2 + 4 = 6

By observing the sign changes, we can determine the intervals where the function changes sign:

-3 to -2: The function changes sign (positive to negative)
-2 to -1: The function does not change sign
-1 to 0: The function changes sign (negative to positive)
0 to 1: The function does not change sign

Step 3: Determine the interval for the real zeros
From the sign changes above, we can infer that there is at least one real zero located between -2 and -1. Therefore, the correct answer is option D: -1 and 0.

Note: It's important to bear in mind that this method provides an estimate based on the observed sign changes, and the exact number and location of the real zeros may require further analysis or calculations.