Answer:
The company will break even at x=1 and x=4.
Step-by-step explanation:
The given profit function is
[tex]P(x)=-x^4-2x^3+21x^2+22x-40[/tex]
Where, x is number of commercials.
Use synthetic division or long division to find the factors of the function.
[tex]P(x)=(x-1)(x^3+3x^2-18x-40)[/tex]
[tex]P(x)=(x-1)(x+2)(x^2+x-20)[/tex]
[tex]P(x) = -(x - 4) (x - 1) (x + 2) (x + 5)[/tex]
Equate P(x)=0, to find the break even points.
[tex]-(x - 4) (x - 1) (x + 2) (x + 5)=0[/tex]
Use zero product property and equate each factor equal to zero.
[tex]x=4,1,-2,-5[/tex]
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,
[tex]P(x)\rightarrow -\infty \text{ as }x\rightarrow \infty[/tex]
[tex]P(x)\rightarrow -\infty \text{ as }x\rightarrow -\infty[/tex]
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.
