Math 289 โ Winter 2007 โ Problem Set 6
Due February 21, 2007
1.(10) Solve the system
x5+y5= 33 , x +y= 3 .
2.(10) Find the real solutions of the equation
4
โ97 โx+4
โx= 5 .
3.(10) Let P(x) be a polynomial of degree 2007, so that P(k) = k
k+1 for k= 0,1,2, . . . , 2007. Find
the value P(2008).
4.(15)
(a) Let P(x) be a nonzero quadratic polynomial, P(โ1) 6= 0, P(0) 6= 0, P(1) 6= 0. Let
๎d
dx ๎2๎P(x)
x3โx๎=f2(x)
g2(x)
for polynomials f2(x) and g2(x). Find the smallest possible degree of f2(x).
(b) Let P(x) be a nonzero polynomial of degree less than 2006, P(โ1) 6= 0, P(0) 6= 0, P(1) 6= 0.
Let
๎d
dx ๎2006 ๎P(x)
x3โx๎=f2006(x)
g2006(x)
for polynomials f2006(x) and g2006 (x). Find the smallest possible degree of f2006(x).
5.(15) Prove that if P(x), Q(x), R(x) and S(x) are polynomials so that
P(x5) + xQ(x5) + x2R(x5) = (x4+x3+x2+x+ 1)S(x),
then xโ1 is a factor of P(x).
6.(25) Prove that every function of the form
f(x) = a0
2+ cos x+
N
X
n=2
ancos(nx)
with |a0|<1, has positive as well as negative values in the period [0,2ฯ). Also, prove that the
function
F(x) =
100
X
n=1
cos ๎n3
2x๎
has at least 40 zeros in the interval (0,1000).