  Assignment 69

 "In the judgment of most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day generation of mathematicians." - (Albert Einstein) Emmy Noether (1882-1935): Making considerable contributions to modern abstract algebra, the German-born Noether was a primary investigator of the structure of noncommunitative algebras. She developed much of the modern theory of mathematical rings and ideals. Overcoming both prejudice against women and the rise of Adolph Hitler in Germany (Noether was Jewish), she became an outstanding teacher at Bryn Mawr College and at the Institute of Advanced Study at Princeton. Her unexpected death in 1935 as a result of an operation to remove a tumor shocked the mathematical world. How did Herkimer keep insects out of his haunted house? Answer: He put up scream doors. Herky wants to know: If people from Poland are called Poles, whey aren't people from Holland called Holes? If someone offers you a penny for your thoughts and you put in your two cents worth, what happens to the other penny? ASSIGNMENT #69 Reading: Section 9.6, pages 568-570. Written: Page 571/5-13. Mathematical word analysis:THEOREM: From the Greek word theoros (a site, something to look at). To ancient mathematicians, theorems (statements that could be proved on the basis of other statements) were worthy of observation.
Here are some examples demonstrating solution processes in solving rational equations:

Example 1: Solve 3/2 + 1/x = 2.

Solution: 3/2 + 1/x = 2

==> 1/x = 1/2

==> x = 2. Check: 3/2 + 1/2 = 2. OK.

Example 2: Solve 3/x + x = 4.

Solution: 3/x + x = 4

==> x(3/x + x) = 4x

==> 3 + x2 = 4x

==> x2 - 4x + 3 = 0

==> (x-3)(x-1) = 0

==> x = 3 or x = 1. Check the answers. Both work.

Example 3: Solve 3x/(x-6) = 5 + 18/(x-6).

Solution: 3x/(x-6) = 5 + 18/(x-6)

==> (x-6)[3x/(x-6)] = (x-6)[5 + 18/(x-6)]

==> 3x = 5(x-6) + 18

==> 3x = 5x - 30 + 18

==> -2x = -12

==> x = 6. Checking, we note that x = 6 does not work. The equation has no solution.

 Problem: Solve 1/x = 4/x3. Solution (with communication): 1/x = 4/x3 ==> x3 = 4x ==> x(x2-4) = 0 ==> x(x+2)(x-2) = 0 ==> x = -2 or x = 0 or x = 2. Checking, we note that x = 0 does not satisfy the original equation. The solution set if {-2, 2}. Problem: Solve a/(x-b) = c/d for x.   Solution (with communication): a/(x-b) = c/d ==> c(x-b) = ad ==> x-b = ad/c (assuming c is not zero) ==> x = ad/c + b = (ad+bc)/c