By Bob Miller

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Structure and Representation of Jordan Algebras

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5, −3, −1, 1, 3, 5, 7, . . The difference between consecutive odds is also 2. We do the even and odd integer problems exxxxactly the same way. x = odd integer (the smallest); x + 2 = next consecutive odd integer (the middle); x + 4 = largest. x2 = square of the smallest; one more than the largest = x + 4 + 1; product (answer in multiplication) of the smallest and the middle = x(x + 2). x2 + x + 4 + 1 = x(x + 2) x2 + x + 5 = x2 + 2x x2’s cancel x + 5 = 2x x=5 The answers are 5, 7, and 9. As I said, most students like this kind of problem.

Sometimes you have to read and reread the problems. EXAMPLE 2— In Springfield, the election for town mayor was close. The winner received 20 more votes than the loser. If 4800 votes were cast, how many votes did each get? In a problem like this, you try to let x equal the smaller number (or the smallest if there are more than two). The loser received x votes. The winner received 20 more, or x + 20. If you add the winner and loser, you get 4800 votes: x + x + 20 = 4800. Solving, we get x = 2390.

Divide each side by the whole coefficient of x, including the sign. x = 4 The root is x = 4. To check, substitute x = 4 in the original equation— always, always the original in case of error. 4(4) + 8 ՘ 10(4) − 16 24 = 24 Yes, it is correct!!!! It is not necessary to memorize these steps. You should just keep them in front of you. With practice you will know them. Not only that, but you will be able to skip some steps and change the order. You will know without anyone telling you. Also do each problem in the book before you do new ones.

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