Final answer:
The polynomial 18x^3 - 120x^2 - 40x can be factorized by first taking out the common factor 6x, and then factoring the remaining quadratic. The factored form is 6x(5)(3x - 1)(x - 7), which is option C.
Step-by-step explanation:
The question asks for the completely factored form of the polynomial 18x^3 - 120x^2 - 40x. To factor this polynomial, we will first look for a common factor in all terms. We can see that 6x can be factored out, which gives us:
6x(3x^2 - 20x - 20)
Now, we need to factor the quadratic expression 3x^2 - 20x - 20. By looking for two numbers that multiply to (3)(-20) = -60 and add to -20, we find that -25 and 5 fulfill this condition. So, we rewrite the middle term -20x as -25x + 5x and factor by grouping.
6x(3x^2 - 25x + 5x - 20)
Grouping the terms, we get:
6x((3x^2 - 25x) + (5x - 20))
Factor further:
6x(x(3x - 25) + 5(3x - 1))
Since (3x - 25) and (3x - 1) are not the same, we cannot factor them as a common factor. However, we can divide 25 by 5, which gives us:
6x(5x(3x - 5) + 5(3x - 1))
Now it is clear that we have a common factor of 5:
6x * 5(x(3x - 5) + (3x - 1))
Which simplifies to:
6x(5)(3x - 1)(x - 7), This is the factored form, which corresponds to option C.