Polynomials (Table of contents)


Problem 1

A monic polynomial \(f(x)\) of fourth degree satisfies \(f(1)=10\), \(f(2)=20\) and \(f(3)=30\). Determine \(f(12)+f(-8)\).

Problem 2

Consider complex polynomials \(P(x)=x^n+a_1x^{n-1}+\cdots +a_n\) with the zeros \(x_1,\dots,x_n\), and \(Q(x)=x^n+b_1x^{n-1}+\cdots+ b_n\) with the zeros \(x_1^2,\dots,x_n^2\). Prove that if \(a_1+a_3+a_5+\cdots\) and \(a_2+a_4+a_6+\cdots\) are real numbers, then \(b_1+b_2+\dots+b_n\) is also real.

Problem 3

If a polynomial \(P\) with real coefficients satisfies for all \(x\) \[P(\cos x)=P(\sin x),\] show that there exists a polynomial \(Q\) such that \(P(x)=Q(x^4-x^2)\) for each \(x\).

Problem 4

  • (a) Prove that for each \(n\in\mathbb{N}\) there is a polynomial \(T_n\) with integer coefficients and the leading coefficient \(2^{n-1}\) such that \(T_n(\cos x)=\cos nx\) for all \(x\).

  • (b) Prove that the polynomials \(T_n\) satisfy \(T_{m+n}+T_{m-n}= 2T_mT_n\) for all \(m,n\in\mathbb{N}\), \(m\geq n\).

  • (c) Prove that the polynomial \(U_n\) given by \(U_n(2x)=2T_n(x)\) also has integer coefficients and satisfies \(U_n(x+x^{-1})=x^n+x^{-n}\).

The polynomials \(T_n(x)\) are known as the Chebyshev polynomials.

Problem 5

Prove that if \(\cos\frac pq\pi=a\) is a rational number for some \(p,q\in\mathbb{Z}\), then \(a\in\{0,\pm\frac12,\pm1\}\).

Problem 6

Prove that the maximum in absolute value of any monic real polynomial of \(n\)-th degree on \([-1,1]\) is not less than \(\frac1{2^{n-1}}\).

Problem 7

The polynomial \(P\) of \(n\)-th degree is such that, for each \(i=0,1,\dots,n\), \(P(i)\) equals the remainder of \(i\) modulo 2. Evaluate \(P(n+1)\).

Problem 8

A polynomial \(P(x)\) of \(n\)-th degree satisfies \(P(i)=\frac1i\) for \(i=1,2,\dots,n+1\). Find \(P(n+2)\).

Problem 9

Let \(P(x)\) be a real polynomial.

  • (a)If \(P(x)\geq0\) for all \(x\), show that there exist real polynomials \(A(x)\) and \(B(x)\) such that \(P(x)=A(x)^2+B(x)^2\).

  • (b) If \(P(x)\geq0\) for all \(x\geq0\), show that there exist real polynomials \(A(x)\) and \(B(x)\) such that \(P(x)=A(x)^2+xB(x)^2\).

Problem 10

Prove that if the equation \(Q(x)=ax^2+(c-b)x+(e-d)=0\) has real roots greater than 1, where \(a,b,c,d,e\in\mathbb{R}\), then the equation \(P(x)=ax^4+bx^3+cx^2+dx+e=0\) has at least one real root.

Problem 11

A monic polynomial \(P\) with real coefficients satisfies \(|P(i)| < 1\). Prove that there is a root \(z=a+bi\) of \(P\) such that \((a^2+b^2+1)^2 < 4b^2+1\).

Problem 12

For what real values of \(a\) does there exist a rational function \(f(x)\) that satisfies \(f(x^2)=f(x)^2-a\)? (A rational function is a quotient of two polynomials.)

Problem 13

Find all polynomials \(P\) satisfying \(P(x^2+1)= P(x)^2+1\) for all \(x\).

Problem 14

Find all \(P\) for which \(P(x)^2-2=2P(2x^2-1)\).

Problem 15

If the polynomials \(P\) and \(Q\) each have a real root and \[P(1+x+Q(x)^2)=Q(1+x+P(x)^2),\] prove that \(P\equiv Q\).

Problem 16 (IMO04.2)

Find all polynomials \(P(x)\) with real coefficients satisfying the equality \[P(a-b)+P(b-c)+P(c-a)=2P(a+b+c)\] for all triples \((a,b,c)\) of real numbers such that \(ab+bc+ca=0\).

Problem 17

A sequence of integers \((a_n)_{n=1}^{\infty}\) has the property that \(m-n\mid a_m-a_n\) for any distinct \(m,n\in\mathbb{N}\). Suppose that there is a polynomial \(P(x)\) such that \(|a_n| < P(n)\) for all \(n\). Show that there exists a polynomial \(Q(x)\) such that \(a_n= Q(n)\) for all \(n\).

Problem 18 (IMO06.5)

Let \(P(x)\) be a polynomial of degree \(n > 1\) with integer coefficients and let \(k\) be a natural number. Consider the polynomial \(Q(x)=P(P(\dots P(P(x))\dots))\), where \(P\) is applied \(k\) times. Prove that there exist at most \(n\) integers \(t\) such that \(Q(t)=t\).

Problem 19

If \(P\) and \(Q\) are monic polynomials such that \(P(P(x))=Q(Q(x))\), prove that \(P\equiv Q\).

Problem 20

Let \(m,n\) and \(a\) be natural numbers and \(p < a-1\) a prime number. Prove that the polynomial \(f(x)=x^m(x-a)^n+p\) is irreducible.

Problem 21

Prove that the polynomial \(F(x)=(x^2+x)^{2^n}+1\) is irreducible for all \(n\in\mathbb{N}\).

Problem 22

A polynomial \(P(x)\) has the property that for every \(y\in \mathbb{Q}\) there exists \(x\in\mathbb{Q}\) such that \(P(x)=y\). Prove that \(P\) is a linear polynomial.

Problem 23

Let \(P(x)\) be a monic polynomial of degree \(n\) whose zeros are \(i-1,i-2,\dots,i-n\) (where \(i^2=-1\)) and let \(R(x)\) and \(S(x)\) be the real polynomials such that \(P(x)=R(x)+iS(x)\). Prove that the polynomial \(R(x)\) has \(n\) real zeros.

Problem 24

Let \(a,b,c\) be natural numbers. Prove that if there exist coprime polynomials \(P,Q,R\) with complex coefficients such that \[P^a+Q^b=R^c,\] then \(\frac1a+\frac1b+\frac1c > 1\).

Corollary: The Last Fermat Theorem for polynomials.

Problem 25

Suppose that all zeros of a monic polynomial \(P(x)\) with integer coefficients are of module 1. Prove that there are only finitely many such polynomials of any given degree; hence show that all its zeros are actually roots of unity, i.e. \(P(x)\mid(x^n-1)^k\) for some natural \(n,k\).