Question

The volume in cubic feet of a box can be expressed as (x)=x^3-6x^2=8x , or as the product of three linear factors with integer coefficients. The width of the box is x-2.

Factor the polynomial to find linear expressions for the height and the length. Show your work.

x^3-6x^2+8x
x(x^2-6x+8)
x(2-x)(14x)

is this right ?

Answer 1

I think you mean x^3 - 6x^2 + 8x for the equation written in the problem.
No, the answer you got isn't quite right. The first factoring is correct though.

To factor x^3 - 6x^2 + 8x, the first obvious factor here is x, since all terms in the expression contain x. Thus,
x(x^2 - 6x + 8)
You got this one right as I see above. Then, to factor x^2 - 6x + 8, what we'll do is first list all factors of 8 (the constant in the expression), and from those factors, we choose the pair that when added, will result to -6 (the numerical coefficient of x in the expression):
1 x 8 || 2 x 4 || -1 x -8 || -2 x -4
From these, we can see that the factors which has a sum of -6 are -2 and -4. Therefore, we can now factor the expression as
x(x^2 - 6x + 8)
= x(x - 2)(x - 4)

Since it was stated in the problem that the factor x - 2 is the width, therefore, the length and height are x and x - 4.
Factoring isn't really that hard. Just keep on practicing on these kind of problems, and if you feel you're confident or good enough on factoring, you won't need to list the factors like we did here - just one look at the expression and you'll know the factors. ;)

Hope this helps~ `u`