- Set the profit function \(p(x) = x^3 - 4x^2 + 5x - 20\) to zero to find the break-even year.
- Factor the expression by grouping to get \((x^2 + 5)(x - 4) = 0\).
- Identify that \(x^2 + 5 = 0\) has no real solutions, while \(x - 4 = 0\) yields \(x = 4\).
- Conclude that the break-even point occurs in year 4, so the answer is \(\boxed{4}\).
Explanation
- Problem Analysis
The entrepreneur's profit function is defined by the polynomial \(p(x) = x^3 - 4x^2 + 5x - 20\), where \(p(x)\) represents the profit in hundreds of dollars and \(x\) represents the number of years the company has been in operation. To "break even" means the total profit is exactly zero, which requires us to solve the equation:
\[x^3 - 4x^2 + 5x - 20 = 0.\]
We are looking for an integer solution for \(x\) from the given set of options: 1, 2, 3, or 4.
- Factoring the Profit Function
We can solve the cubic equation \(x^3 - 4x^2 + 5x - 20 = 0\) by using the method of factoring by grouping. We group the terms of the polynomial as follows:
\[(x^3 - 4x^2) + (5x - 20) = 0.\]
Now, we factor out the common terms from each group:
- From the first group \((x^3 - 4x^2)\), we can factor out \(x^2\), giving \(x^2(x - 4)\).
- From the second group \((5x - 20)\), we can factor out 5, giving \(5(x - 4)\).
Substituting these back into the equation, we get:
\[x^2(x - 4) + 5(x - 4) = 0.\]
Now, notice that \((x - 4)\) is a common binomial factor. We can factor it out:
\[(x^2 + 5)(x - 4) = 0.\]
- Solving for the Break-Even Year
To find the roots of the equation \((x^2 + 5)(x - 4) = 0\), we set each factor equal to zero:
- \(x^2 + 5 = 0 \implies x^2 = -5\). This equation has no real solutions because the square of a real number cannot be negative.
- \(x - 4 = 0 \implies x = 4\).
The only real root for the equation is \(x = 4\). Thus, the entrepreneur breaks even in year 4.
- Conclusion
Based on the analysis, the entrepreneur breaks even when \(x = 4\). This matches one of the provided options. Therefore, the entrepreneur will break even in year 4.
The final answer is \(\boxed{4}\).
Examples
Understanding break-even analysis is essential for any business owner. For example, if a bakery invests in a new oven that costs a certain amount, they need to calculate how many units of bread must be sold to cover that cost. By representing profit as a function of time or sales volume, as shown in this problem, business owners can determine the exact point where revenue exceeds costs, helping them plan for long-term growth and financial stability.