Mathematics 467
Homework (due Feb. 20)
A. Hulpke
14∗ ) (Partial Fractions)
a) Let f , g, h ∈ Q[x] with a = deg(g), b = deg(h) and a + b > deg( f ). Show that if gcd(g, h) = 1,
then there are r, s ∈ Q[x] such that
f (x) = r(x)g(x) + s(x)h(x)
and deg r < b and deg h < a. (Hint: If 1 = c ⋅ g + d ⋅ h, divide f ⋅ c by h with remainder.)
b) Let r, p ∈ Q[x]. Show that we can write
r(x) = s1 (x)p l (x) + s2 (x)p l−1 (x) + ⋯ + s l−1 (x)p2 (x) + s l (x)p(x) + s l+1 (x)
with s i ∈ Q[x] and deg s i (x) < deg p(x).
c) Let f , g1 , g2 , . . . , g k ∈ Q[x], gcd(g i , g j ) = 1 for i =/ j and 1 ≤ e i ∈ Z such that deg f < ∑i e i ⋅ deg g i .
Show that we can write (partial fraction decomposition, as done in calculus):
g1e1 (x) ⋅
f (x)
g2e2 (x)⋯g ke k (x)
s1,e (x)
s1,1 (x) s1,2 (x)
+ 2
+ ⋯ + e11
g1 (x) g1 (x)
g1 (x)
s2,1 (x)
s2,e (x)
+
+ ⋯ e22
+⋯
g2 (x)
g2 (x)
s k,e (x)
s k,1 (x)
+
+ ⋯ e kk
g k (x)
g k (x)
=
with deg s i, j < deg g i .
15) Let f (x), g(x) ∈ Q[x]. Suppose that α, β ∈ C such that f (α) = 0, g(β) = 0.
a) Show that Res( f (x − y), g(y), y) has a root α + β. (Note: Similar expressions exist for α − β, α ⋅ β
and α/β.)
√ √
b) Construct a polynomial f ∈ Q[x] such that f ( 5 + 3 2) = 0.
16) Let F be a field. For a polynomial
n
f = ∑ a i x i ∈ F[x]
i=0
we define the derivative D f as
n
D f = ∑ i ⋅ a i x i−1
i=1
(with i being the the corresponding sum of 1s.)
a) Show that for f , g ∈ F[x] we have that D( f g) = (D f )g + f (Dg).
b) Suppose that f = a n ∏(x − α i ) in F. Show that f has repeated roots if and only if f and D f have
a nonconstant gcd.
17) The discriminant of a polynomial f ∈ F[x] of degree m with leading coefficient a is defined as
disc( f ) = (−1)
m(m−1)
2
Res( f , D f )/a
with D f defined as in problem 16.
a) Show that if f = ∏i (x = α i ), we have that disc( f ) = ∏i< j (α i − α j )2 .
b) Show that disc( f ) = 0 if and only if f has multiple roots (possibly in a larger field than F).
18) Consider a (2-dimensional) robot arm as
depicted.
We want to find out the angles θ 1 , θ 2 to which
the joints have to be set to move the hand to
coordinates (a, b). For simplification, assume
the two arms have length l1 = 3, l2 = 4. To avoid
using trigonometric functions, set s i = sin(θ i ),
c i = cos(θ i ) (i = 1, 2). Then s 2i + c 2i = 1.
Write down equations that determine a, b in
terms of the variables c1 , s1 , c2 , s2 . (You will have
to use the formulas for sin(α + β) and cos(α +
β).)
Problems marked with a ∗ are bonus problems for extra credit.
y
θ2
l2
(a,b)
l1
θ1
x