Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

HERENDI, T. and PETHŐ, A., Trinomials, which are divisible by quadratic polynomials

Trinomials, which are divisible by quadratic polynomials 63 Let consider the sum: I — II + III: F

n(x) • F

k(x) - (x) -F

k(x) + (~l)

k • x

k • F

n_k_i (x) = F

n(x) • F

k.i(x) - F

n^(x) • F

k-t{x) - (-1)* • x'1 • F

n.

k(x)+ x • (.F

n(x) • F

k_

2(x) -F

n_ 1(x) • F

k.

x(x) + {-l)

k • x

k~

2 • F

n_

k+

l(x)) . The right hand side of this équation is equal to zéro by the induction hypothesis, so the equality holds for k too. Lemma 2. (a) The polynomial F

n{x) has degree [y] and its roots are — ? where 1 < j < [y] and £

n+i is a n + 1-th primitive root of unity; (b) F

n(x) has a rational root if and only if gcd(n + 1,12) > 3. PROOF. (a) By définition we have Fo(x) = F\(x) = 1, so deg(Fo(x)) = [|] and deg(

JFi(a:)) = [|]. Let n> 2 and suppose that deg (F

k(x)) = if k < n. It is easy to see that the leading coeffitient of F

k(x) is positive. So deg {F

n(x)) = deg (fn-i(s) + x • F

n_

2(x)) = " fi " = max(deg(

JF

n_i(x)),deg(

JF

n_2(x)) + 1) = . Let be a récurrence sequence with the définition: u

m = r • u

m_i + 5 • Um-2, where r, s ^ 0, r

2 + 45 ^ 0 and \u

0\ + |wi| > 0. Then u

m = a • a

m + b • ß

m(m = 0,1, 2,. . .), where a, ß is the two différent roots of the polynomial z

2 - r • z - s and a = U

of_~

a

U

l, 6 = U

l

^

9

a

a (see e. g. [2]). Let suppose now that t is a root of F

n(x) and define {u

m}™

=

0 ^y ^e following recurrence: Uq = U\ = 1 and u

m := w

m_i + t • u

m-2 if m > 2. It is clear that F

m(t) = u

m (

m = 0,1,2,...), and if t ^ then v/T+lí-l /l-0T+4í\

m V

/TT4í + 1 /1 + VT+4Í u

m = _ ^ • + 2VT+4

t V 2 / 2VTT4Í V 2 / / - n—:—TT\ "i+l / „ rz 7— \ m-fl" 1 l ( 1 + y/1 + 4A / 1 - y/1 + 4* VI + 41 IV 2 By the choice of t we have 0 = F

n(t) = u

n which means i + VïTTt\

n+

l /î-x/ITïP^

1 -0, 1. e.