find the real zeros of the polynomial:

P(x) = 24x^3 - 76x^2 + 2

so we are solving

24x^3 - 76x^2 + 2 = 0
dividing by 2 helps a bit

12x^3 - 38x^2 + 1 = 0

after a few trials with ±1, ±1/2, ±1/3 I found that 1/6 works, so (6x-1) is a factor.
Using long division gave me
12x^3 - 38x^2 + 1 = (6x-1)(2x^2 - 6x - 1)

I will leave it with you to find the 2 zeros you get from solving the quadratic
2x^2 - 6x - 1 = 0

(hint, one of them is appr. 3.158)

Thank you I just think I got overwhelmed!

To find the real zeros of a polynomial, we need to solve the equation P(x) = 0. In this case, the polynomial is P(x) = 24x^3 - 76x^2 + 2.

Here's how you can find the real zeros:

Step 1: Factor out the greatest common factor, if possible. In this case, there is no common factor other than 1.

Step 2: Use the Rational Root Theorem to list all possible rational roots of the polynomial. The Rational Root Theorem states that any rational root of the polynomial P(x) = 0 will be of the form p/q, where p is a factor of the constant term (in this case, 2) and q is a factor of the leading coefficient (in this case, 24). Therefore, the possible rational roots are ±1, ±2, ±(1/2), and ±(1/3).

Step 3: Check each possible root using synthetic division or long division to see if it indeed satisfies the equation P(x) = 0. Start with the first possible root and continue until you find a root that satisfies the equation.

Let's start with the possible root x = 1:

Using synthetic division, we have:

1 | 24 -76 0 2
|____ 24 -52 -52
| 24 -52 -52 -50

The remainder is -50, which means that x = 1 is not a root of the polynomial.

Now let's try x = -1:

-1 | 24 -76 0 2
|_____ -24 100 -100
| 24 -100 100 -98

The remainder is -98, which means that x = -1 is not a root of the polynomial.

We can continue this process for the remaining possible rational roots (±2, ±(1/2), ±(1/3)) until we find a root that satisfies the equation P(x) = 0.

In this case, the real zeros of the polynomial P(x) = 24x^3 - 76x^2 + 2 are not rational numbers. The zeros might be irrational or complex numbers. To find these zeros, you would need to use numerical methods such as using a graphing calculator or computer software.