# Problem 723

The profit function for a product is given by P(x) = -n3 +2n2 + 1600n -3000, where x is the number of units produced and sold and P is in hundreds of dollars. Use factoring by grouping to find the numbers of units that will give a profit of 20,000.

Solution:-

Because P(x) is the profit function in hundreds of dollars, a profits of 20,000 is represented by P(x) = 200. So solve the equation 200 = -n3 +2n2 + 1600n -3000

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Get 0 on the left side of the equation.

200 = -n3 +2n2 + 1600n -3000

0 = -n3 +2n2 + 1600n -3200

This equation appears to have  a form that permits factoring by grouping.

Separate the terms into two groups, each having a common factor.

0 = (-n3  + 2n2)  +(1600n – 3200)

Factor –n2 from the first group, then 1600 from the second group.

0 = -n2 (n – 2) + 1600(n – 2)

The common binomial factor is n – 0.

Factor the common factor from each of the two terms.

0 = (n – 2)(1600 – n2)

Notice, 1600 – n2 is a difference of square and can be factored.

0 = (n – 2)(40-n)(40+n)

The negative value cannot represent x, the number of units product, so numbers of units that will give a profit of 20,000 are 2 and 40.