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 ?
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`
To factor the polynomial x^3 - 6x^2 + 8x, we first look for any common factors among the terms. In this case, x is a common factor. By factoring out x, we get:
x(x^2 - 6x + 8)
Now, we need to factor the quadratic expression x^2 - 6x + 8. To do this, we look for two integers that multiply to give 8 and add to give -6. The integers -2 and -4 satisfy these conditions because -2 * -4 = 8 and -2 + (-4) = -6. Therefore, we can factor the quadratic expression as:
(x - 2)(x - 4)
Now, we have the factorization in the form:
x(x - 2)(x - 4)
According to the question, the width of the box is x - 2. So, the height and length would be the remaining factors, which are x and x - 4, respectively.
Therefore, the linear expressions for the height and length are x and x - 4, respectively.
So, your factorization is correct:
x^3 - 6x^2 + 8x = x(x - 2)(x - 4)
No, your factorization is not correct. Let's go through the correct factorization step-by-step:
Given the volume of the box as (x) = x^3 - 6x^2 + 8x.
Step 1: We start by factoring out the greatest common factor, which is x:
(x) = x(x^2 - 6x + 8).
Step 2: Next, we need to factor the quadratic expression inside the parentheses, x^2 - 6x + 8. To do this, we need to find two numbers that multiply to give 8 and add up to -6. The numbers -2 and -4 fit this criteria, so we can factor the quadratic expression as:
x^2 - 6x + 8 = (x - 2)(x - 4).
Thus, the complete factorization is:
(x) = x(x - 2)(x - 4).
Now we can assign the linear expressions for the height and length:
Height = x - 2,
Length = x - 4.
Hence, the correct factorization is x(x - 2)(x - 4), and the linear expressions for the height and length are (x - 2) and (x - 4) respectively.