Question

As a financial consultant, the CFO of a major television company has shared the profit function P(x) = −x4 - 2x3 + 21x2 + 22x − 40 with you which models the profit of the company based on the number of commercials , x, they air in a 24-hour period. DESCRIBE the appearance of the graph to the CFO. Use complete sentences, and focus on the end behaviors of the graph and how to use the function P(x) to find where the company will break even (where P(x) = 0).

Answer 1

The company will break even at x=1 and x=4.

Explanation:

The given profit function is

`P(x)=-x^4-2x^3+21x^2+22x-40`

Where, x is number of commercials.

Use synthetic division or long division to find the factors of the function.

`P(x)=(x-1)(x^3+3x^2-18x-40)`

`P(x)=(x-1)(x+2)(x^2+x-20)`

`P(x) = -(x - 4) (x - 1) (x + 2) (x + 5)`

Equate P(x)=0, to find the break even points.

`-(x - 4) (x - 1) (x + 2) (x + 5)=0`

Use zero product property and equate each factor equal to zero.

`x=4,1,-2,-5`

Therefore the zeros of the function P(x) are 4, 1, -2 and -5. Since x is number of commercials, therefore the value of x must be positive. Therefore the company will break even at x=1 and x=4.

The higher degree of the function is even and the leading coefficient is negative. So,

`P(x)\rightarrow -\infty \text{ as }x\rightarrow \infty`

`P(x)\rightarrow -\infty \text{ as }x\rightarrow -\infty`

Since the value of x must be positive, therefore the left end point is y-intercept of the graph, i.e., (0,-40).

Therefore the company faces huge loss when the number of commercials increases unboundly.

The graph of the function is given below.