A polynomial $\ f(x)$ has integer coefficients such that $\ f(0)$ and $\ f(1)$ are both odd numbers. Prove that $\ f(x) = 0$ has no integer solutions.
58927
Integer coefficient polynomial $p(x)$ has no integer roots of $\,p(0)$ and $p(1)$ are odd [Parity Root Test]
30860
Prove that $f(x)=0$ in $[-1,1]$.
148813
Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
7199
Suppose that $P(x)$ is a polynomial of degree $n$ such that $P(k)=\dfrac{k}{k+1}$ for $k=0,1,\ldots,n$. Find the value of $P(n+1)$
30511
Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]
6722
to show $f$ is a polynomial in $z$
13147
Proof verification - Minimal polynomial of $\alpha$ has integer coef.s iff its an alg. integer
155575
Proving a polynomial $f(x)$ composite for infinitely many $x$
107133
Let f(x) be a polynomial satisfying $\lim_{x \to\infty} \frac{x^4f(x)}{x^8+1}=3,$ $f(2)=5 $ and $f(3)=10$, $f(-1)=2$ and $f(-6)=37$
134342
Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit.
35478
Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
116307
Polynomial whose only values are squares
48873
Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.
154360
Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.
94682
Polynomial: prove there is no double root
27494
$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$