query: Let $f(x)=7x^{32}+5x^{22}+3x^{12}+x^2$. Find the remainder when $x^2+1$ divides $f(x)$ and $xf(x)$.
positive_passages.docid: 142769
positive_passages.text: Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.
negative_passages.docid: 138164
negative_passages.text: If the range of $f(x)= \frac{x^2+x+c}{x^2+2x+c}, x\in \mathbb R$ is $\left[\frac 56, \frac 32\right]$ then what is $c$?
negative_passages.docid: 36906
negative_passages.text: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
negative_passages.docid: 113789
negative_passages.text: Given that$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+...+a_1x+a_0$, find $a_7+a_6+a_5+a_4+...+a_1+a_0$
negative_passages.docid: 48053
negative_passages.text: Show that 13 divides $2^{70}+3^{70}$
negative_passages.docid: 132093
negative_passages.text: Solve $f(f(x))=f(x)$
negative_passages.docid: 13248
negative_passages.text: How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
negative_passages.docid: 113506
negative_passages.text: negative number divided by positive number, what would be remainder?
negative_passages.docid: 21439
negative_passages.text: For which $k\in \Bbb N$ does $x^2+x+1$ divide $ (x + 1)^{2k}+x^{2k} + 1$?
negative_passages.docid: 124295
negative_passages.text: Divison and remainder in Prolog
negative_passages.docid: 4475
negative_passages.text: Find the remainder when $2^{2016}$ is divided by $47$
negative_passages.docid: 63750
negative_passages.text: let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$ then find $f(x)$
negative_passages.docid: 91914
negative_passages.text: Prove that for the Fibonacci sequence $(F_n)$, $F_n$ divides $F_{2n}$.
negative_passages.docid: 100336
negative_passages.text: Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
negative_passages.docid: 111493
negative_passages.text: Division algorithm for polynomials in R[x], where R is a commutative ring with unity.
negative_passages.docid: 143392
negative_passages.text: How to calculate the division and remainder of a big number manually
negative_passages.docid: 48873
negative_passages.text: 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 $n!$.