Polynomial Equations

If $x, y, z$ are all real numbers and $x^2+y^2+z^2=2\left(x-y-z\right)-3$, then the value of $2x-3y+4z$ is:

If $a^2+b^2+c^2=2\left(a-b-c\right)-3$, then the value of $\left(a+b+c\right)$ is:

Let, $f\left(x\right)=x^n+5x^{n-1}+3,n>1$ is an integer. Prove that $f\left(x\right)$ cannot be expressed as a product of two polynomials, each of which has all its coefficient integers and degree at least $1$ 

Prove that $16x^3-35x^2+105x+175$ is irreducible over $Z$.

Show that $x^4+x^3-x+1$ is irreducible over $Z$.

Let $P\left(x\right)$ be a polynomial such that $x·P(x-1)=(x-4)P\left(x\right)\forall x\in R$ Find all such $P\left(x\right)$.

What is Diophantine equation?

State and prove Gauss’s Lemma theorem.

Find the integral roots of the polynomial $f\left(x\right)=x^3+6x^2+11x+6$

A polynomial of odd degree with real coefficients must have -

If $x-3$ and $x+3$ are the factors of $4x^3+a{x}^2+bx$, then the values of $a$ and $b$ are respectively -

If $f\left(x\right)=2x^3+m{x}^2-13x+n$ and $2,3$ are roots of the equation $f\left(x\right)=0$, then the values of $m$ and $n$ are

Let $P\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots +a_n{x}^n$ be a polynomial in which $a_i$ is a non-negative integer for each $i\in \left(0,1,2,3,\dots ,n\right)$. If $P\left(1\right)=4$ and $P\left(5\right)=136$, then $P\left(3\right)$ is

The zeros of a polynomial depends on the

What Is Meant by Zeros of Polynomial?

The number of real roots of the equation $\frac{{\left(x^2+1\right)}^3}{x^3}+\frac{x^2+1}{3x}=0(x\ne 0)$ is

The polynomial $9$ has no zero.

Zero of the polynomial $3$ is $1$.

Find the zero of the polynomial: $3$

The zero of the polynomial $2$ is $\frac{1}{2}$.