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Foundations of Secondary School Mathematics
December 5, 2001
Show all your work on these pages! Total score = 100
- (30 points)
- (20) Apply the division algorithm to express
- 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.
- (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.
- (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