To find the function that represents the length of the shipping box, we can use the relationship between the volume \( v \), width \( w \), height \( h \), and length \( l \) of the rectangular box, given by:
\[ v(x) = l(x) \cdot w(x) \cdot h(x) \]
From the problem, we have:
- Volume function: \( v(x) = 2x^3 + x^2 - 25x + 12 \)
- Width function: \( w(x) = x - 3 \)
- Height function: \( h(x) = x + 4 \)
We can express the length function in terms of the volume, width, and height:
\[ l(x) = \frac{v(x)}{w(x) \cdot h(x)} \]
Now, we can calculate \( w(x) \cdot h(x) \):
\[ w(x) \cdot h(x) = (x - 3)(x + 4) \]
Using the distributive property:
\[ w(x) \cdot h(x) = x^2 + 4x - 3x - 12 = x^2 + x - 12 \]
Next, we will find \( l(x) \):
\[ l(x) = \frac{2x^3 + x^2 - 25x + 12}{x^2 + x - 12} \]
Now we need to perform polynomial long division to divide \( 2x^3 + x^2 - 25x + 12 \) by \( x^2 + x - 12 \).
- Divide \( 2x^3 \) by \( x^2 \) to get \( 2x \).
- Multiply \( 2x \) by \( x^2 + x - 12 \) to get \( 2x^3 + 2x^2 - 24x \).
- Subtract \( (2x^3 + 2x^2 - 24x) \) from \( (2x^3 + x^2 - 25x + 12) \):
\[ (2x^3 + x^2 - 25x + 12) - (2x^3 + 2x^2 - 24x) = (x^2 - 2x^2) + (-25x + 24x) + 12 = -x^2 - x + 12 \]
- Now, divide \( -x^2 \) by \( x^2 \) to get \( -1 \).
- Multiply \( -1 \) by \( x^2 + x - 12 \) to get \( -x^2 - x + 12 \).
- Subtract:
\[ (-x^2 - x + 12) - (-x^2 - x + 12) = 0 \]
So, we can conclude:
\[ l(x) = 2x - 1 \]
Thus, the function that represents the length of the shipping box is:
\[ \boxed{l(x) = 2x - 1} \]