Assignment #1
Date Due: Thursday, Sept. 10, 2009
Section 0 numbers 3, 5, 9, 16, 18
and
a) Let R[x] denote the set of all polynomials in the variable xwith real number coeffi-
cients, i.e. R[x]=!anxn+an−1xn−1+· · · +a0|ai∈Rand n∈N"
Ex: 5x2−17
√10 x+ 2 and πx107 −15 are polynomials (sums of real numbers times non-
negative interger powers of x).
Prove that every polynomial in R[x] of positive degree is either irreducible itself or is
a product of irreducible polynomials in R[x].
(An irreducible polynomial h(x) is one that cannot be factored into a product of
polynomials of strictly lower degree.)
b) Let pbe a prime integer:
i) Prove that one cannot find nonzero positive integers aand bsucht that a2=pb2.
Hint: Use a proof by contradiction: suppose there are such intergers aand b. Explain
why one may assume aand bto be relatively prime. (Recall: two positive integers are
relatively prime if they have no common factors other than 1). Then proceed to show
that p|aand p|b.
ii) Deduce from (i) that √pis irrational, i.e., √p$∈ Q, if pis prime.
c) Compute each of the following:
i) i27 ii) (1 −i)(2 + i)
(1 −2i)(1 + i)iii) |3−4i|iv) #i0
1−i$4
v) #i0
1−i$−1
d) Express the following in polar-coordinate form z=|z|(cos θ+isin θ).
i) z=iii) z= 7 iii) z= 1 + √3i