To find the height of the box given its volume, length, and width, we will use the relationship:
[tex]\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \][/tex]
In this case, the volume [tex]\( f(x) \)[/tex] is [tex]\( x^3 - x^2 - 17x - 15 \)[/tex]. The length is [tex]\( (x-5) \)[/tex] and the width is [tex]\( (x+1) \)[/tex]. Let's denote the height by [tex]\( h(x) \)[/tex].
1. First, express the volume in terms of length, width, and height:
[tex]\[ f(x) = (x - 5)(x + 1)h(x) \][/tex]
2. Substitute the given expression for [tex]\( f(x) \)[/tex]:
[tex]\[ x^3 - x^2 - 17x - 15 = (x - 5)(x + 1)h(x) \][/tex]
3. To find [tex]\( h(x) \)[/tex], we need to solve for [tex]\( h(x) \)[/tex]. Divide both sides of the equation by the product of the length and width:
[tex]\[ h(x) = \frac{x^3 - x^2 - 17x - 15}{(x - 5)(x + 1)} \][/tex]
4. Next, we will perform polynomial long division to divide [tex]\( x^3 - x^2 - 17x - 15 \)[/tex] by [tex]\( (x - 5)(x + 1) \)[/tex].
Let’s find the product [tex]\( (x - 5)(x + 1) \)[/tex]:
[tex]\[ (x - 5)(x + 1) = x^2 + x - 5x - 5 = x^2 - 4x - 5 \][/tex]
Now, we need to divide [tex]\( x^3 - x^2 - 17x - 15 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex].
Perform polynomial long division:
1. [tex]\( x^3 \div x^2 = x \)[/tex]
2. Multiply [tex]\( x \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:
[tex]\[ x \cdot (x^2 - 4x - 5) = x^3 - 4x^2 - 5x \][/tex]
3. Subtract this product from the original polynomial:
[tex]\[ (x^3 - x^2 - 17x - 15) - (x^3 - 4x^2 - 5x) = 3x^2 - 12x - 15 \][/tex]
4. Now divide the new polynomial [tex]\( 3x^2 - 12x - 15 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:
[tex]\[ 3x^2 \div x^2 = 3 \][/tex]
5. Multiply [tex]\( 3 \)[/tex] by [tex]\( x^2 - 4x - 5 \)[/tex]:
[tex]\[ 3 \cdot (x^2 - 4x - 5) = 3x^2 - 12x - 15 \][/tex]
6. Subtract this result from [tex]\( 3x^2 - 12x - 15 \)[/tex]:
[tex]\[ (3x^2 - 12x - 15) - (3x^2 - 12x - 15) = 0 \][/tex]
The division results in no remainder, so:
[tex]\[ h(x) = x + 3 \][/tex]
Thus, the height of the box is represented by the expression:
[tex]\[ x + 3 \][/tex]
