## Real roots of a polynomial

Cách đây vài tháng tôi có đưa lên blog này một số bài toán về nghiệm thực của đa thức, cụ thể ở các link sau:

Dưới đây là file pdf tổng hợp các bài trên sau khi nhận được sự góp ý từ các bạn đồng nghiệp và các em học sinh.

## Lagrange’s interpolation polynomial (4)

Problem 17. Prove that, for infinitely many positive integers $n$, there exists a polynomial $P$ of degree $n$ with real coefficients such that $P(1),P(2),\cdots, P(n+2)$ are different whole powers of $2$.

Problem 18. Suppose $q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:

(i)  $m-n \,$ divides $q_{m}-q_{n}$ for $m > n \geq 0,$

(ii) there is a polynomial $P$ such that $|q_{n}| < P(n) \,$ for all $n$

Prove that there is a polynomial $Q$ such that $q_{n}= Q(n)$ for all $n$.

Problem 19. Let $P\in\mathbb{R}[x]$ such that for infty of integer number $c$ : Equation $P(x)=c$ has more than one integer root. Prove that $P(x)=Q((x-a)^{2})$, where $a\in\mathbb R$ and $Q$ is a polynomial.

Problem 20. Find all the polynomials $P(x)$ with odd degree such that

$P(x^{2}-2)=P^{2}(x)-2.$

Problem 21. Suppose $p(x)$ is a polynomial  with integer coefficients assumes at $n$ distinct integral values of $x$ that are different form $0$ and in absolute value less than  $\dfrac{(n-[\frac n2])!}{2^{n-[\frac n2]}} .$ Prove that $p(x)$ is irreducible.

Prove that the bound may be replaced by $(\dfrac d2)^{n-[\frac n2]}(n-[\frac n2])!$ where $d$ is minimum distance between any two of the $n$ integral values of $x$ where $p(x)$ assumes the integral values considered.

## Lagrange’s interpolation polynomial (3)

Problem 9. Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of  $\displaystyle\frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}}$  is an integer for each $k = 0,1, ..., m$.

Problem 10. Let $f\left ( x \right )=x^{n}+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+...+a_{1}x+a_{0}$ be a polynomial. Prove that we have an $\displaystyle i\in \left \{ 1,2,...,n \right \}\mid \left | f\left ( i \right ) \right |\geq \frac{n!}{\binom{n}{i}}$.

Problem 11.  Let $(F_n)_{n\geq 1}$ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying

$P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.$ Prove that $P(1983) = F_{1983} - 1.$

## Lagrange’s interpolation polynomial (2)

Problem 1. Let $P$ be a polynomial of degree at most $n$ satisfying $\displaystyle P(k)=\frac{1}{C^k_{n+1}}\,\,\forall k=\overline{0,n}.$ Determine $P(n+1)$.

Problem 2. A polynomial $P(x)$ has degree at most $2k$, where $k = 0, 1,2,\cdots$. Given that for an integer $i$, the inequality $-k \le i \le k$ implies $|P(i)| \le 1$, prove that for all real numbers $x$, with $-k \le x \le k$, the following inequality holds $|P(x)| \leq 2^{2k}.$

Problems 3. Prove that at least one of the numbers $|f(1)|,|f(2)|,\cdots,$ $|f(n+1)|$ is greater than or equal to $\dfrac{n!}{2n}.$ Where

$f(x) = x^n + a_1x^{n-1} + \cdots+ a_n\,\, (\quad a_i \in \mathbb R, \quad i = 1, \ldots , n,n\in\mathbb{N}.)$

Problem 4. Prove that any monic polynomial (a polynomial with leading coefficient $1$) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

Problem 5. Let $p$ be a prime number and $f$ an integer polynomial of degree $d$ such that $f(0) = 0,f(1) = 1$ and $f(n)$ is congruent to $0$ or $1$ modulo $p$ for every integer $n$. Prove that $d\geq p - 1$.

Problem 6. Let $P$ be a polynomial of degree $n\in\mathbb{N}$ satisfying $P(k)=2^k\,\,\forall k=\overline{0,n}.$ Prove that $P(n+1)=2^{n+1}-1$.

Problem 7.  $P(x)$ is a polynomial of degree $3n\,\, (n\in\mathbb{N})$ such that

$P(0) = P(3) = \cdots = P(3n) = 2,\,\,\, P(1) = P(4) = \cdots = P(3n-2) = 1,$

$P(2) = P(5) = \cdots = P(3n-1) = 0, \quad\text{and}\quad P(3n+1) = 730.$

Determine $n$.