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.