Answer:
We are given the profit function [tex]P(x)=-x^{4} -2x^{3}+21x^{2} +22x-40[/tex], where x is the number of commercials aired in 24- hour period.
Now, as [tex]P(x)=-x^{4} -2x^{3}+21x^{2} +22x-40[/tex].
Using long division gives us that this polynomial can be factorized as,
[tex]P(x)=(x-1) \times (x^{3}+3x^{2}-18x-40)[/tex]
i.e. [tex]P(x)=(x-1) \times (x+2) \times (x^{2}+x-20)[/tex]
i.e. [tex]P(x)=-(x-1) \times (x+2) \times (x-4) \times (x+5)[/tex].
Since we want to find where will the company break.
So, we equate P(x) = 0.
i.e. [tex]-(x-1) \times (x+2) \times (x-4) \times (x+5) = 0[/tex].
i.e. (x-1) = 0, (x+2) = 0, (x-4) = 0 and (x+5) = 0.
i.e. x = 1, x = -2, x = 4 and x = -5.
Since, x represents the number of commercials. Therefore, it cannot have negative values.
Thus, x = 1 and x = 4.
This implies that the company will break even the number of commercials is 1 and 4.
Graphically, as the degree of the polynomial is 4 .i.e even and the leading co-efficient is -1 i.e. negative, this gives us that the function P(x) will increase at the start and will decrease in the end.
Moreover, we can see from the graph below that,
[tex]P(x)\rightarrow -\infty[/tex] as [tex]x\rightarrow \infty[/tex] and [tex]P(x)\rightarrow -\infty[/tex] as [tex]x\rightarrow -\infty[/tex].
Hence, the company will face huge loss as the number of commercials increases without any bounds.
Further, as the value of x is always positive i.e.x ≥ 0. The y-intercept is when x = 0 i.e. P(0) = -40 i.e. y-intercept is at ( 0,-40 ).
Also, we can see that since the roots of P(x) are not repeating, P(x) cuts x-axis at 4 points namely (-5,0), (-2,0), (1,0) and (4,0).
