I have a fair variety of problems that I am having some issues with--most I have worked out but need confirmation on, others I have no real idea what to do with.

The ones I need todouble checkare:

(1)

Assuming that none of the solutions are undefined,simplify each of the following:

(a) $\displaystyle \frac{a^2 b^2 c}{abc}$

My answer: Either abc or ab(I'm not too sure which, though I am leaning towards "ab")

(b) $\displaystyle \frac{a^{2m} b^{m+1}}{a^m b}$

My answer: $\displaystyle a^{2m-m}b^m$ or $\displaystyle a^mb^m$

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(2)

Simplify:

(a)$\displaystyle \frac{x-y}{x+y} + \frac{x+y}{x-y}$

My answer: $\displaystyle \frac{x-y}{x+y} + \frac{x+y}{x-y} \rightarrow \frac{(x-y)(x-y)}{(x+y)(x-y)} + \frac{(x+y)(x+y)}{(x+y)(x-y)}$ $\displaystyle \rightarrow \frac{(x-y)^2}{(x+y)(x-y)} + \frac{(x+y)^2}{(x+y)(x-y)} \rightarrow \frac{(x-y)^2+(x+y)^2}{(x+y)(x-y)} \rightarrow (x-y)+(x-y)$

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(3)

If you arehfeet above the earth's surface, the distance,d, that you can see is approximatelyd(in miles)= $\displaystyle \sqrt{\frac{3h}{2}}$.

Suppose you are on the roof of a building 900 ft. tall. how far can you see?(calculate to two decimal places)

My answer: $\displaystyle h=900$ so $\displaystyle \sqrt{\frac{3h}{2}}=d \rightarrow \sqrt{\frac{3*900}{2}}=d \rightarrow \sqrt{\frac{2700}{2}}=d \rightarrow \sqrt{1250}=d \ \therefore \ d=36.74$ Thus, I can see for 36.74 Mi.

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(4)

According to Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots?

(1) $\displaystyle f(a)=a^5-4a^2-7$

My answer:

(a):1

(b):0

(2) $\displaystyle f(x)=3x^3+9x^2+8x$

My answer:

(a):0

(b):1

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Now for the ones I have no idea about.

(1)

Solve the equation $\displaystyle x^3+9x^2+x+2=0$, if -2 is a root.

(I have tried to solve this one but when I do synthetic division on it I get the equation: $\displaystyle x^2+x=0$ which I don't think will work)

(2)

Find the dimensions of a rectangleawith the greatest area whose perimeter is 30 feet.

(I got nothing--I'm totally confused as to what to do with this one.)

Thank you!