1. Find the dimensions of a rectangle "a" with the greatest area whose perimeter is 30 feet.

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P = 2l +2w

30 = 2l + 2w

15 = l + w

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Area = l(w)

A = w(15-w)

A = 15w - w^2

This is a quadratic with a=-1,b=15,c= -A

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Greatest area occurs when w = -b/2a = -15/(2(-1)) = 7.5 feet (this is the width)

Since L+w = 15, length = 7.5 ft

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2. Given x^3-4x^2+2x+1=0

# of sign changes in f(x) = 2

# of sign changes in f(-x) = 1

(a) How many possible positive roots are there?: 0 or 2

(b) How many possible negative roots are there?: 1

(c) What are the possible rational roots? 1 and -1

(d) Using synthetic substitution, which of the possible rational roots is actually a root of the equation? +1

(e) Find the irrational roots of the equation. (Hint: Use the quadratic formula to solve the depressed equation.)

1)....1....-4....2....1

-------1-----3....-1..|..0

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Quotient: x^2-3x-1

Roots: x = [3 +- sqrt(9 - 4*1*-1)]/2

x = [3 +- sqrt(13)]/2