Final answer:
The factored form of the given function y = -2x^3 - 2x^2 + 112x is y = 2x(x - 8)(x + 7). By expanding and simplifying the factored form, it can be checked that it matches the given function.
Step-by-step explanation:
The given function is y = -2x^3 - 2x^2 + 112x.
To write the function in factored form, we can start by factoring out the greatest common factor. In this case, the greatest common factor is 2x, so we can rewrite the function as:
y = 2x(-x^2 - x + 56)
Next, we can factor the quadratic expression -x^2 - x + 56. We can use factoring or the quadratic formula to find the factors, but in this case, the expression does not easily factor, so we can use the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(-1)(56))) / (2(-1))
x = (1 ± √(1 + 224)) / 2
x = (1 ± √225) / 2
x = (1 ± 15) / 2
So the solutions to the quadratic equation are x = (1 + 15) / 2 = 8 and x = (1 - 15) / 2 = -7.
Therefore, the factored form of the function is y = 2x(x - 8)(x + 7).
To check by multiplication, we can distribute the factors and simplify:
y = 2x(x - 8)(x + 7)
y = 2x(x^2 - x - 8x + 56)
y = 2x(x^2 - 9x + 56)
y = 2x^3 - 18x^2 + 112x
The factored form matches the given function, so it is correct.