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Polynomial Modeling: Shoe Manufacturer Profit Analysis

Learn how to model a shoe manufacturer's profit using polynomials and find the optimal production quantity for maximum profit.

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Polynomial Modeling: Shoe Manufacturer Profit Analysis

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  1. 3 1 –2 –23 60 3 3 –60 1 1 –20 0 EXAMPLE 5 Standardized Test Practice SOLUTION Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division.

  2. The correct answer is A. ANSWER EXAMPLE 5 Standardized Test Practice Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x3– 2x2– 23x + 60 = (x – 3)(x2 + x – 20) = (x – 3)(x + 5)(x – 4) The zeros are3, –5, and4.

  3. EXAMPLE 6 Use a polynomial model BUSINESS The profit P(in millions of dollars) for a shoe manufacturer can be modeled by P = –21x3 + 46x where xis the number of shoes produced (in millions). The company now produces 1 million shoes and makes a profit of $25,000,000, but would like to cut back production. What lesser number of shoes could the company produce and still make the same profit?

  4. 1 21 0 –46 25 21 21 –25 21 21 –25 0 EXAMPLE 6 Use a polynomial model SOLUTION Substitute25forPinP = –21x3 + 46x. 25 = –21x3 + 46x 0 = 21x3– 46x + 25 Write in standard form. You know that x = 1 is one solution of the equation. This implies that x – 1 is a factor of 21x3– 46x + 25. Use synthetic division to find the other factors.

  5. ANSWER The company could still make the same profit producing about 700,000 shoes. EXAMPLE 6 Use a polynomial model So,(x – 1)(21x2 + 21x – 25) = 0. Use the quadratic formula to find that x 0.7 is the other positive solution.

  6. What if? In Example 6, how does the answer change if the profit for the shoe manufacturer is modeled by P =–15x3 + 40x? 9. for Examples 5 and 6 GUIDED PRACTICE Find the other zeros of fgiven that f (–2) = 0. 7. f (x) = x3 + 2x2– 9x – 18 8. f (x) = x3 + 8x2 + 5x – 14 ANSWER 3and–3 ANSWER 1 and–7 ANSWER The company could still make the same profit producing about 900,000 shoes.

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