Q-1) Let \(N\) be a positive integer such that \(N+1\) is prime. Choose \(a_i \in \left \{ 0,1 \right \}\) for \(i=0,..,N\), such that the \(a_i\) are not all equal. Also let \(f(x)\) be a polynomial such that \(f(i)=a_i\) for all \(i=0,..,N\). Show that the degree of \(f(x)\) is at least \(N\).
Q-2) Let \(a,b\) be distinct integers. Show that the polynomial \((x-a)^2(x-b)^2+1\) is irreducible over \(\mathbb{Z}\).
Q-3) Let \(n\) be a positive integer and let \(P_n\) denote the set of integer polynomials of the form \(a_0+a_1x+…+a_nx^n\), where \(|a_i|\le 2\) for \(i=0,..,n\). Find for each natural number \(k\), the number of elements of the set \(A_n(k)=\left \{ f(k):f \in P_n \right \}\).
Q-4) Find all \(x\) such that \(\left \{ (x+1)^3 \right \}=x^3\) where \(\left \{ x \right \}\) denotes the fractional part of \(x\).
Q-5) Let \(p(x)=x^2+ax+b\) be a quadratic polynomial over \(\mathbb{Z}\). Show that for any integer \(n\), there exists an integer \(M\) such that \(p(n)p(n+1)=p(M)\).
Q-6) Given any four distinct positive real numbers, show that one can choose three numbers \(A,B,C\) from among them, such that all three quadratic equations
have real roots, or all 3 of them have imaginary roots.
Q-7) Let \(k,n\) and \(r\) be positive integers.
(a) Let \(Q(x)=x^k+a_1x^{k+1}+…a_nx^{k+n}\) be a polynomial with real coefficients.
a) Show that the function \(\frac{Q(x)}{x^k}\) is strictly positive for all real \(x\) satisfying
(b) Let \(P(x)=b_0+b_1x+..b_rx^r\) be a non zero polynomial with real coefficients. Let \(m\) be the smallest number such that \(b_m \ne 0\). Prove that the graph of \(y=P(x)\) cuts the X-Axis at the origin (i.e. \(P\) changes signs at \(x=0\)) if and only if \(m\) is an odd integer.
Q-8) If \(P(x)=x^n+a_1x^{n-1}+..a_n\) be a polynomial with real coefficients and \(a_1^2<a_2\) then prove that not all roots of \(P(x)\) are real.
Q-9) Consider the polynomial \(ax^3+bx^2+cx+d\) where \(a,b,c,d\) are integers such that \(ad\) is odd and \(bc\) is even. Prove that not all of its roots are rational.
Q-10) Suppose \(a\) and \(b\) are two positive real numbers such that the roots of the cubic equation \(x^3-ax+b=0\) are all real. If \(\alpha\) is a root of this cubic with minimal absolute value, prove that
<<In case there is any confusion over any example (solved or not), problem or concept, please feel free to create a thread in the appropriate forum. Our memebrs/experts will try their best to help solve the problem for you>>