I need to find the critical number of 4x^3-36x^2+96x-64 by factoring. Basically I need to know the zeros.

If you plug in x=1, you find that f(x)=4x^3-36x^2+96x-64 evaluates to zero. From here, use either synthetic division or long division to get the quadratic 4x^2-32x+64. Reduce to x^2-8x+16 and solve by factoring into (x-4)^2.

So 4x^3-36x^2+96x-64=4(x-1)(x-4)^2.

The critical numbers are the values for x in which the function has a horizontal tangent line, so where the first derivative is zero.

F1(x)=12x^2-72x+96
0=12x^2-72x+96
0= x^2-6x+8
0=(x-4)(x-2)
The critical numbers are
x=4 and x=2

If you need to find the zeros of a cubic by factoring then the resault is:

4(x-4)^2-(x-1), so the zeros are at 4, and 1, and these are also the critical points.

You did not specify if 4x^3-36x^2+96x-64 was your function, or the first derivative of the function, however if it is not then you should take the first derivative which should leave you a quadratic function, then use the quadratic formula on the remaining second-degree polynomial.

Well, everything divides by 4, so you can take that out for a start, leaving

x^3 - 9x^2 + 24x - 16

We need three numbers with a product of -16 (1, 2, 4, 8 and 16 are the only possibilities) that sum to -9.

Shouldn't be too hard to figure out from there.

To find the critical numbers or zeros of the function f(x) = 4x^3 - 36x^2 + 96x - 64, we can start by looking for common factors and using the Rational Zero Theorem.

Step 1: Factor out the common factor, if any:
In this case, we can see that 4 is a common factor. So, we can rewrite the function as:
f(x) = 4(x^3 - 9x^2 + 24x - 16).

Step 2: Determine the possible rational zeros using the Rational Zero Theorem:
According to the Rational Zero Theorem, the possible rational zeros of the function are all possible ratios of the factors of the constant term (in this case, 16) divided by the factors of the leading coefficient (in this case, 4).
So, the possible rational zeros are: ±1, ±2, ±4, ±8, ±16.

Step 3: Test the possible rational zeros using synthetic division:
Checking each of the possible rational zeros in the synthetic division will help us identify the actual zeros of the function.

Let's start with x = 1:
Performing synthetic division with the coefficients (1, -9, 24, -16) and dividing by (1, -1, 1) yields:
1 | 1 -9 24 -16
|______1 -8 16
1 -8 16 0
As the remainder is 0, it means that (x - 1) is a factor of the function.

Now, we can factor out the result of the synthetic division:
f(x) = 4(x^3 - 9x^2 + 24x - 16) = 4(x - 1)(x^2 - 8x + 16).

Step 4: Find the remaining zeros by factoring the remaining quadratic:
To find the zeros of the quadratic factor (x^2 - 8x + 16), we can either factor it or use the quadratic formula. In this case, we can factor as follows:
(x^2 - 8x + 16) = (x - 4) (x - 4),
which means the quadratic factor has a repeated root x = 4.

Therefore, the critical numbers (or zeros) of the function f(x) = 4x^3 - 36x^2 + 96x - 64 are x = 1 (a single root) and x = 4 (a repeated root).