# Algebra

Please show work for each!

54. Find the polynomial function of degree 3 with real coefficients that satisfies the given conditions.

Zero of 4 having multiplicity 2 and zero of 2 having multiplicity 1; f(1) = -18

48. Use the intermediate value theorem for polynomials. 1 and 2

58. Show that the real zeros of each polynomial function satisfy the given conditions.

no real zero is greater than 1

14. Solve each variation problem.

If m varies jointly as z and p, and m = 10 when z = 2 and p =7.5, find m when z = 5 and p = 7.

28. The current in a simple electrical circuit varies inversely as the resistance. If the current is 50amps when the resistance is 10 Ohms, find the current if the resistance is 5ohms.

40. The number the long distance phone calls between two cities in a certain time period varies directly as the populations p1 and p2 of the cities, and inversely as the distance between them. If 10,000 calls are made between two cities 500 mi apart, having populations of 50,000 and 125,000, find the number of calls between two cities 800mi apart, having populations of 20,000 and 80,000.

52. Concept check, Work each problem.

What happens to y if y is inversely proportional to x, and x is tripled?

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#### Solution Preview

54. Find the polynomial function of degree 3 with real coefficients that satisfies the given conditions.

Zero of 4 having multiplicity 2 and zero of 2 having multiplicity 1; f(1) = -18

Let the function be f(x) = k(x - a)(x - b)(x - c)

a = b = 4 and c = 2; Also, f(1) = -18

-18 = k(1 - 4)(1 - 4)(1 - 2) = -9k

k = 2

f(x) = 2(x - 4)^2 (x - 2)

f(x) = 2x^3 - 20x^2 + 64x - 64

48. Use the intermediate value theorem for polynomials. 1 and 2

f'(x) = 6x - 1 and f'(c) = 6c - 1

f'(x) = [f(2) - f(1)]/(2 - 1) = (6 + 2)/1 = ...

#### Solution Summary

The expert finds the polynomial functions of degree 3 with real coefficients that satisfies specific conditions. A complete, neat and step-by-step solutions are provided in the attached file.