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 71 2. p- is one of the rational roots of F

n-\(x) . As A ^ 0 by Lemma 4 (a) a\A , so in case 1 ( k is also even ) A-a - (b

n~

2 mod 2 • fn—2 {a, b

2) - B • b

k~

2 mod 2 fk-2 (M

2)) is finitely many linear équation for the possible B -s. In case 2, as by Lemma 2, _F

n_i (#) has at most three différent rational roots and similarly to the previous case a\ A , we have only finitely many a, b pairs which satisfies the necessary conditions. By Lemma 3 if /

n_i (a,6

2) = 0 then fn—2 [a-,b

2) / 0 so we have again finitely many linear équations for the possible B -s. Proof of Theorem 2. (a) Let a, 6, A G Z such that x

2 - bx - a | x

n - Bx

k - A. As B / 0 similarly to Theorem 1 (b) b

k~

l mod 2 • (a,b

2) ± 0. By Lemma 4 B • b

k~

l mod 2 • /*_! (a, 6

2) = mod 2 • /

n_! (a, 6

2) . If we suppose that 6 = 0 then the équation is a polynomial équation for a, which has only finitely many solution in a so the number of possible values for A is also finite ( and efFectively determinable ). Let suppose now that 6^0. Then B • F

k-î(p-) fon — k As deg(JW) = and deg(f

n_i) ' n —

1 ], there exist real numbers Mi, M2, £1, £2 s

o that if x > £1 then < Mi • \x\^ 2 ^ and if x > x

2 then |jP

n_!(a;)| > M

2 • {xy^ (M

x

, M

2 > 0). Let x

0 = max (1, xi

, x

2) and suppose that |p-| > XQ then B - Mi b

2 il" i — k I > B-Fk-i(û) n—k I — Fn_l > M 2 M As n — k > 4 and > 1 we get B • Mi M

2 > B • Mi M

2 • |6

n~ > > It means that there exists a constant Mo > 0 so that —Mo < p- < Mo for ail the possible a, 6 G Z pairs. Hence there exists M > 0 so that \Fk-i (p") | < M for ail the possible a, 6 pairs. Or which is the same, (il) B-M 16 n — k I > F

n, -