# 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 This solution is **FREE** courtesy of BrainMass!

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 = 8

6c - 1 = 8

c = 1.5

We see that 1 < 1.5 < 2

The IV Theorem is verified.

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

No real zero is greater than 1

We divide f(x) synthetically by 1.

1 2 -1 2 -2 4 -4

2 1 3 1 5

2 1 3 1 5 1

Since all the numbers in the last row are positive, x = 1 is an upper bound for the zeros of f(x). Thus, no real zero of f(x) 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.

m = kzp

k = m/zp = 10/(2 * 7.5) = 2/3

m = (2/3)zp

m = (2/3)(5 * 7) = 70/3

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.

I = k/R

k = IR = 50 * 10 = 500

I = 500/R

I = 500/5 = 100 A

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.

n = k p1 p2 / d

k = n d/(p1 p2) = 10000 * 500/(50000 * 125000) = 8 * 10^-4

n = (8 * 10^-4) p1 p2 / d

n = (8 * 10^-4)(20000 * 80000)/800 = 1600

52. Concept check, Work each problem.

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

y = k/x

When x is tripled, it becomes 3x and y becomes y/3.

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