To determine the average profit per unit of the product sold, we need to follow these steps:
1. Define the profit expression: The profit earned from selling a quantity of [tex]\( x+2 \)[/tex] units of a product is given by:
[tex]\[ P(x) = x^4 - x^3 - 24 \][/tex]
2. Define the quantity expression: The quantity of products sold is:
[tex]\[ Q(x) = x + 2 \][/tex]
3. Calculate the average profit per unit: To find the average profit per unit, we need to divide the profit expression by the quantity expression:
[tex]\[ \text{Average Profit per Unit} = \frac{P(x)}{Q(x)} = \frac{x^4 - x^3 - 24}{x + 2} \][/tex]
4. Simplify the expression: We must simplify the fraction to find out which of the given options matches the simplified expression.
We perform polynomial long division of [tex]\( x^4 - x^3 - 24 \)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ \frac{x^4 - x^3 - 24}{x + 2} \][/tex]
Step 1: Divide the leading term:
[tex]\[ \frac{x^4}{x} = x^3 \][/tex]
Step 2: Multiply [tex]\( x^3 \)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ x^3(x + 2) = x^4 + 2x^3 \][/tex]
Step 3: Subtract [tex]\( x^4 + 2x^3 \)[/tex] from [tex]\( x^4 - x^3 - 24 \)[/tex]:
[tex]\[ (x^4 - x^3 - 24) - (x^4 + 2x^3) = -3x^3 - 24 \][/tex]
Step 4: Repeat the process with [tex]\(-3x^3 - 24\)[/tex]. Divide by [tex]\( x \)[/tex]:
[tex]\[ \frac{-3x^3}{x} = -3x^2 \][/tex]
Step 5: Multiply [tex]\(-3x^2 \)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ -3x^2(x + 2) = -3x^3 - 6x^2 \][/tex]
Step 6: Subtract [tex]\( -3x^3 - 6x^2 \)[/tex] from [tex]\( -3x^3 - 24 \)[/tex]:
[tex]\[ (-3x^3 - 24) - (-3x^3 - 6x^2) = 6x^2 - 24 \][/tex]
Step 7: Repeat the process with [tex]\( 6x^2 - 24 \)[/tex]. Divide by [tex]\( x \)[/tex]:
[tex]\[ \frac{6x^2}{x} = 6x \][/tex]
Step 8: Multiply [tex]\( 6x \)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ 6x(x + 2) = 6x^2 + 12x \][/tex]
Step 9: Subtract [tex]\( 6x^2 + 12x \)[/tex] from [tex]\( 6x^2 - 24 \)[/tex]:
[tex]\[ (6x^2 - 24) - (6x^2 + 12x) = -12x - 24 \][/tex]
Step 10: Repeat the process with [tex]\(-12x - 24 \)[/tex]. Divide by [tex]\( x \)[/tex]:
[tex]\[ \frac{-12x}{x} = -12 \][/tex]
Step 11: Multiply [tex]\(-12\)[/tex] by [tex]\( x + 2 \)[/tex]:
[tex]\[ -12(x + 2) = -12x - 24 \][/tex]
Step 12: Subtract [tex]\( -12x - 24 \)[/tex] from [tex]\( -12x - 24 \)[/tex]:
[tex]\[ (-12x - 24) - (-12x - 24) = 0 \][/tex]
Thus, the simplified expression is:
[tex]\[ x^3 - 3x^2 + 6x - 12 \][/tex]
Hence, the average profit per unit of product sold is:
[tex]\[ B. \ x^3 - 3 x^2 + 6 x - 12 \][/tex]
