1. x^4+bx^3+cx^2+dx+e=0 (where b, c, d, and e are real and rational) has solutio

Question

1. x^4+bx^3+cx^2+dx+e=0 (where b, c, d, and e are real and rational) has solutions 2, 3, and -5i, find the value of e/b

2. If x+y=6 and xy=7, then find the value of x^3+y^3

3. If cos@=x and sin@<0, then find tan@ in terms of x.
@= theta

4. John wants to enclose a rectangular field for his goats. He has 1200 feet of fence and a river that will form one of the four sides of the field. What is the maximum area in square feet that he can enclose?

5. find the product of the real roots of the equation: 2x^4+5x^3+x^2+20x-28=0

Explanation / Answer

1. Since imaginary roots must appear in conjugate pairs, 5i is also a root. Then the equation is (x-2)(x-3)(x-5i)(x+5i)=0 ---> (x2-5x+6)(x2+25)=0 ---> x4-5x3+31x2-125x+150=0. So e/b = 150/-5 = -30.

2. x3+y3 = (x+y)(x2-xy+y2) = (x+y)((x+y)2-3(xy)) = (6)((6)2-3(7)) = 90.

3. sin@ = -(1-x2) ---> tan@ = -(1-x2)/x.

4. P = 1200 = L+2W.   A = LW = (1200-2W)W.

Then set the derivative equal to 0 to find maximum:

A' = 0 = 1200-4W ---> W=300. Then L=600 and A=600*300 = 180000.

5. If you factor this equation you get (x2+4)(2x+7)(x-1)=0, so the roots are 2i, -2i, -3.5, and 1. The product of the real roots is then -3.5 * 1 = -3.5.

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