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 Problem: 1 2 3 Total Score:

# Midterm 2

### December 5, 2001

Show all your work on these pages! Total score = 100

1. (30 points)
• (20) Apply the division algorithm to express
x4 - 7 x3 + x2 -3 x + 1
as a polynomial multiple of 2x2 + 2 with a remainder a polynomial of degree one or less.
• (10) Write a paragraph explaining the similarities and differences between the division algorithms for polynomials and for numbers, in terms that a high school student could be expected to understand. Be sure to explain why it is essential for the polynomials to have coefficients in a field.

2. (30 points)
• (20) Explain why 5 has a multiplicative inverse modulo 12 but 9 does not. In general, which residue classes modulo n have multiplicative inverses? Explain your answer carefully.
• (10) Integer arithmetic modulo 12 is often called ``clock arithmetic.'' Write a paragraph explaining exactly to what extent this is appropriate. In particular explain what multiplication modulo 12 means in terms of clocks.

3. (40 points)
• (20) Calculate the image of the point (5,4) after rotation by 40o about the point (2,1).
• (20) Let O = (0,0) represent the origin in the cartesian plane, and let P = (a,b) be an arbitrary point. Explain from basic principles why, for any angle K, the two compositions are equal: RK,P o T(a,b) = T(a,b) o RK,O. In words, translating by (a,b) and then rotating K degrees about (a,b) has the same effect as rotating K degrees about the origin and then translating by (a,b). You may give a geometric argument or calculate the effect of the two compositions on a point (x,y).