Question

The profit function for a product is given by π(x) = −x3 + 28x2 − 57x − 450, where x is the number of units produced and sold. If break-even occurs when 6 units are produced and sold:
i. Find the quadratic factor of π(x).
ii. Find a number of units other than 6 that gives break-even for the product.

Answer 1

Thank you for asking this question. Here is your answer:

First of all we would divide the equation by 10.

Which will give us -1 88 180 and the remainder would be 0

and then we will factorize it further:

(- x2 + 88 x + 180) (x-10)

-(x-90) (x+2) (x-10)

If there is any confusion please leave a comment below.

Answer 2

Answer:

1) $x^2+22x-75$

2) x=25,  x=-3

Step-by-step explanation:

Given : $\pi (x) = -x^3 + 28x^2-57x -450$

1) To find the quadratic factor of π(x)

First we factorize the function as 6 unit is produced when break-even happen

Therefore, one of the factor is x=6

Divide  $\pi (x) = -x^3 + 28x^2-57x -450$ with factor (x-6) by long division

Long division - Dividend=divisor×quotient+remainder

Dividend= $-x^3 + 28x^2-57x -450$

Divisor= (x-6)

Dividend/divisor=quotient+remainder

$\frac{-x^3 + 28x^2-57x -450}{(x-6)}= x^2+22x-75 + 0$

Therefore, the quadratic factor is $x^2+22x-75$

2) To find a number of units other than 6 that gives break-even for the product.

Now, we factorize further to get more factors

Using middle term split,

$x^2+22x-75=0$

$x^2-25x+3x-75=0$

$x(x-25)+3(x-25)=0$

$(x-25)(x+3)=0$

x=25, x=-3

Both 25,-3 gives break-even for the product

Hence, $\pi (x) = -x^3 + 28x^2-57x -450=(x-6)(x+3)(x-25)$