select the correct answer. an entrepreneur estimates his total profit (total revenue minus total cost) for…

select the correct answer. an entrepreneur estimates his total profit (total revenue minus total cost) for his proposed company as $p(x)=x^{3}-4x^{2}+5x - 20$, where $p$ is in hundreds of dollars and $x$ is number of years the company has been in business. in which year ($x$) will the entrepreneur break even?\na. year 1\nb. year 2\nc. year 3\nd. year 4

select the correct answer. an entrepreneur estimates his total profit (total revenue minus total cost) for his proposed company as $p(x)=x^{3}-4x^{2}+5x - 20$, where $p$ is in hundreds of dollars and $x$ is number of years the company has been in business. in which year ($x$) will the entrepreneur break even?\na. year 1\nb. year 2\nc. year 3\nd. year 4

Answer

Answer:

D. year 4

Explanation:

Step1: Define break - even condition

At break - even, $p(x)=0$, so we set $x^{3}-4x^{2}+5x - 20 = 0$.

Step2: Factor the polynomial

Group the terms: $(x^{3}-4x^{2})+(5x - 20)=0$. Then $x^{2}(x - 4)+5(x - 4)=0$, which factors to $(x - 4)(x^{2}+5)=0$.

Step3: Solve for x

Set each factor equal to zero. For $x^{2}+5 = 0$, $x^{2}=-5$, and the solutions are $x=\pm\sqrt{-5}$ (non - real). For $x - 4=0$, $x = 4$. So the entrepreneur breaks even in year 4.