Recent questions tagged polynomials

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $P \in \mathbb{F}_{q}[x]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when ...

Consider the set of all possible polynomials$x^3 + a_2x^2 + a_1x + a_0$,with $a_0,a_1,a_2$ integers and |$a_0$| ∈ {0,...,9}, such that for each polynomial, all of its roo...

If $\alpha, \beta, \gamma$ are roots of $x^3 + ax + 1 = 0$, then$ \frac{\alpha }{\beta }+\frac{\beta }{\alpha }+\frac{\beta }{\gamma }+\frac{\gamma }{\beta }+\frac{\alpha...

Consider the equation $x^{7} + 4x^5 + 2x^3 + x + 1 = 0.$ The number of real roots of this equation is:(A) 1 (B) 3 (C) 5 (D) 7

Does there exist a polynomial $q(x)$ with integer coefficients such that $q(1)=2$ and $q(3)=5$ ? Given an example if there is one. Justify, if there is not.

For a polynomial of the form $a_{0}+a_{1} x+\cdots+a_{m} x^{m}$ having degree $m>1$, it is known that the coefficients satisfy the relation $\frac{a_{i-1}}{a_{i-1}+a_{i}}...

The polynomial \( x^4 - ax^3 + bx^2 - cx + 4 \) when divided by \( x - 2 \) leaves a remainder of 5 and when divided by \( x + 2 \) leaves a remainder of 11. Then \( b^2 ...

Let $p(x)$ be a polynomial with real coefficients which satisfies $p(r)=p(-r)$ for every real number $r$. Let $n \geq 5$ be a positive integer. Suppose that $p(i)=i$ for ...

Let $n \geq 100$ be a positive integer. Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, each taking values in the set $\{0,1\}$ such that $\operatornam...