Problem: | 1 | 2 | 3 | Total |

Score: |

- (30 points)
- (20) Apply the division algorithm to express

`x`^{4}- 7 x^{3}+ x^{2}-3 x + 1

as a polynomial multiple of`2x`with a remainder a polynomial of degree one or less.^{2}+ 2 - (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.

- (20) Apply the division algorithm to express
- (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.

- (20) Explain why 5 has a multiplicative inverse modulo 12
but 9 does not. In general, which
residue classes modulo
- (40 points)
- (20) Calculate the image of the point (5,4) after rotation
by 40
^{o}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:`R`. In words, translating by_{K,P}o T_{(a,b)}= T_{(a,b)}o R_{K,O}`(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)`.

- (20) Calculate the image of the point (5,4) after rotation
by 40