To factor the given polynomial expression and match the factors to the dimensions they represent, let's follow a detailed, step-by-step process.
### Given Expression
The volume of the enclosure is given by the polynomial:
[tex]\[ w^3 + 6w^2 - 40w \][/tex]
### Step 1: Simplify the Expression
First, we want to simplify the expression if possible. Notice that each term in the polynomial has a common factor of [tex]\(w\)[/tex], so we can factor [tex]\(w\)[/tex] out:
[tex]\[ w^3 + 6w^2 - 40w = w(w^2 + 6w - 40) \][/tex]
### Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression [tex]\(w^2 + 6w - 40\)[/tex]. We look for two numbers that multiply to [tex]\(-40\)[/tex] (the constant term) and add up to [tex]\(6\)[/tex] (the coefficient of the linear term).
Those two numbers are [tex]\(10\)[/tex] and [tex]\(-4\)[/tex], since:
[tex]\[ 10 \times (-4) = -40 \][/tex]
[tex]\[ 10 + (-4) = 6 \][/tex]
So, the quadratic expression [tex]\(w^2 + 6w - 40\)[/tex] can be factored as:
[tex]\[ w^2 + 6w - 40 = (w + 10)(w - 4) \][/tex]
### Step 3: Combine All Factors
Putting it all together, we get:
[tex]\[ w^3 + 6w^2 - 40w = w(w + 10)(w - 4) \][/tex]
### Final Factored Form
Thus, the factored form of the given polynomial expression is:
[tex]\[ w^3 + 6w^2 - 40w = w(w - 4)(w + 10) \][/tex]
### Matching the Factors to Dimensions
Now we match the factors to the dimensions they represent. Based on typical dimension labels and the fact that one factor should be aligned with the width, length, and height of an enclosure:
- [tex]\(w\)[/tex] represents one dimension (usually the width, [tex]\(w\)[/tex]).
- [tex]\(w - 4\)[/tex] and [tex]\(w + 10\)[/tex] represent the other two dimensions.
Given that the length of the enclosure must be greater than the height:
- [tex]\((w + 10)\)[/tex] should be the length (the largest dimension).
- [tex]\((w - 4)\)[/tex] should be the height (the smallest dimension).
Therefore, matching the factors to dimensions, we get:
- Width ([tex]\(w\)[/tex]): [tex]\(w\)[/tex]
- Height: [tex]\(w - 4\)[/tex]
- Length: [tex]\(w + 10\)[/tex]
So, the complete factorization and matching of the polynomial expression with the corresponding dimensions of the enclosure are:
[tex]\[ w^3 + 6w^2 - 40w = w \times (w - 4) \times (w + 10) \][/tex]
