Question

ou are having a meeting with the CEO of a technology company. You have interpreted the number of laptops produced versus profit as the function P(x) = x4 − 3x3 − 8x2 + 12x + 16. Describe to the CEO what the graph looks like and, in general, how to sketch the graph without using technology. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0).

Answer 1

Answer: We can plot the graph with help of below explanation.

Explanation:

Since, given equation of polynomial,

`P(x) = x^4 - 3x^3 - 8x^2 + 12x + 16`

End behavior : Since, the leading coefficient of the polynomial is positive and even.

Therefore, the end behavior of the polynomial is,

`f(x)\rightarrow -\infty` as
`x\rightarrow -\infty`

And,
`f(x)\rightarrow +\infty` as
`x\rightarrow +\infty`

Points of the curve : since, P(4) = 0

Therefore, (x-4) is the multiple of P(x),

And we can write,
`x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x^3+x^2-4x-4)`

`x^4 - 3x^3 - 8x^2 + 12x + 16=(x-4)(x+1)(x^2-4)`

`x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x+1)(x+2)(x-2)`

Thus, the roots of equation are 4, 2, -1 and -2.

Therefore, x-intercepts of the polynomial are (4,0) (2,0) (-1,0) and (-2,0)

Also, the y-intercept of the polynomial is ( 0,16)

Thus, we can plot the graph with help of the above information.