The volume in cubic feet of a Cd holder can be expressed as V(x)=-x^3-x^2+6x, the depth is 2-x , factor the polynomial to find linear expression for the height and the width. - Do I just factor the -x^3-x^2+6x? It because I know the volume is LWH, so don't I have to subtract the depth from the volume? Thanks

Well, talking about factoring, I always like to bring some pun into it. So, are you ready for a "factoring" fun?!

Let's start by factoring the given polynomial.

V(x) = -x^3 - x^2 + 6x

To factor this, we can first find the common factor:

V(x) = x(-x^2 - x + 6)

Hmm, this quadratic trinomial -x^2 - x + 6 doesn't seem to have a nice factoring pattern, does it?

Don't worry, though! We can rely on the trusty quadratic formula to find its roots. It goes like this: x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = -1, b = -1, and c = 6. So, let's plug those values in:

x = (1 ± √((-1)^2 - 4(-1)(6))) / (2(-1))

Simplifying, we get:

x = (1 ± √(1 + 24)) / (-2)

x = (1 ± √25) / (-2)

x = (1 ± 5) / (-2)

So, the possible values for x are x = -3 and x = 2.

Now, we can plug these values back into the expression for the depth, which is 2 - x, to find the height and width.

For x = -3:
Depth = 2 - (-3) = 2 + 3 = 5, which represents the height or the width.

For x = 2:
Depth = 2 - 2 = 0, which represents the height or the width.

So, the linear expressions for the height and width of the Cd holder can be:
Height = 5 units
Width = 5 units

Remember, these expressions are obtained from the depth, which we subtracted from the volume formula. Voila! Factoring complete, problem solved, and a giggle guaranteed!

To find the linear expressions for the height and width of the CD holder, you need to factor the polynomial expression -x^3-x^2+6x and use the depth 2-x.

To factor the polynomial, you look for common factors first. In this case, there is no common factor other than 1. So, you move on to factoring by grouping or using other techniques like the Rational Root Theorem.

The polynomial -x^3-x^2+6x can be factored as follows:

-x(x^2 + x - 6)

Now, we have a quadratic expression x^2 + x - 6 that can be further factored. This can be factored as follows:

x^2 + x - 6 = (x - 2)(x + 3)

So, the polynomial -x^3-x^2+6x can be factored as:

-x(x - 2)(x + 3)

Now that we have factored the polynomial, we can find the linear expressions for the height and width.

The linear expression for the height is (x - 2) (since the depth is 2 - x).

The linear expression for the width is (x + 3) (since the width has no direct relationship with the depth).

Therefore, the height of the CD holder can be expressed as (x - 2), and the width can be expressed as (x + 3).

Note that to calculate the volume, you do multiply the height, width, and depth: V(x) = (x - 2)(x + 3)(2 - x).

To find the linear expressions for the height and width of the Cd holder, you will indeed need to factor the polynomial -x^3 - x^2 + 6x. However, before factoring, it is important to make a correction to the expression for the volume.

You mentioned that the depth of the Cd holder is given by 2 - x. In the volume formula, the depth should be multiplied by x, not subtracted from it, because the volume is calculated by multiplying the length (which is x in this case) with the width and the height.

Taking this into account, the correct expression for the volume V(x) is:
V(x) = x * (-x^3 - x^2 + 6x)

Now, let's factor the polynomial:
V(x) = x * (-x^3 - x^2 + 6x)
= -x^4 - x^3 + 6x^2

To factor this polynomial, you can start by factoring out the greatest common factor, which is -x^2:
V(x) = -x^2 * (x^2 + x - 6)

Next, factor the quadratic expression inside the parentheses:
V(x) = -x^2 * (x + 3) * (x - 2)

Now that the polynomial has been factored, you can interpret the factors as the dimensions of the Cd holder. According to the volume formula (L * W * H), the width is represented by (x + 3), and the height is represented by (x - 2). Note that the depth is already considered in the previous factoring steps as part of the volume formula.

Therefore, the linear expressions for the height and width of the Cd holder are:
Height: x - 2
Width: x + 3

volume = length x width x height

so
-x^3 - x^2 + 6x = (length x width) x (2-x)
multiply each side by -1

x^3 + x^2 - 6x = (length x width) x (x-2)
do either a long divsion or use synthetic division to get

x^3 + x^2 - 6x = (x-2)(x^2 + 3x)
= (x-2)(x)(x+3)

so the height is (2-x) , that was given
the length is x+3
the width is x

check:
multiply x(2-x)(x+3) to get -x^3 - x^2 + 6x