Certainly! Let's work through each part of the question step-by-step.
1. Factored Form of the Polynomial When [tex]\( P = 4 \)[/tex]:
Given the profit function:
[tex]\[ P = -x^3 + 4x^2 + x \][/tex]
We know that for [tex]\( P = 4 \)[/tex], we can set up the equation and rearrange it to:
[tex]\[ 4 = -x^3 + 4x^2 + x \][/tex]
[tex]\[ -x^3 + 4x^2 + x - 4 = 0 \][/tex]
To find the factored form, we need to factor the polynomial:
[tex]\[ 0 = - (x - 4)(x - 1)(x + 1) \][/tex]
So, the factored form of the polynomial when [tex]\( P = 4 \)[/tex] is:
[tex]\[ -(x - 4)(x - 1)(x + 1) \][/tex]
2. Sales Levels Where Profit is also [tex]\( \$4,000 \)[/tex] (When [tex]\( x = 4 \)[/tex]):
From the factored form:
[tex]\[ -(x - 4)(x - 1)(x + 1) = 0 \][/tex]
We set each factor equal to zero to find the sales levels:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
[tex]\[ x - 1 = 0 \][/tex]
[tex]\[ x = 1 \][/tex]
[tex]\[ x + 1 = 0 \][/tex]
[tex]\[ x = -1 \][/tex]
Since sales level can't be negative, the valid sales levels are:
[tex]\[ \boxed{1 \text{ and } 4} \][/tex]
3. Relative Maximum Profit Between 0 and 5,000 Items Sold:
To find the maximum profit within the range of [tex]\( 0 \)[/tex] to [tex]\( 5 \)[/tex] (in thousands), we analyze the profit function or use technology to plot and find the relative maximum.
The maximum profit is about:
[tex]\[ \boxed{12} \][/tex]
This occurs when the number of items sold is:
[tex]\[ \boxed{3} \][/tex]
So, the complete answers are as follows:
- The factored form of the polynomial when [tex]\( P = 4 \)[/tex] is:
[tex]\[ -(x - 4)(x - 1)(x + 1) \][/tex]
- The sales levels where the profit is also [tex]\( \$4,000 \)[/tex] are:
[tex]\[ 1 \text{ thousand items and } 4 \text{ thousand items sold} \][/tex]
- The maximum profit when the number of items sold is between 0 and 5,000:
[tex]\[
\boxed{12} \text{ thousand dollars when } \boxed{3} \text{ thousand items are sold}
\][/tex]
