Algebraic Equations of Higher Degree

If $\alpha , \beta$ and $\gamma$ are the roots of the equation $x^3-12x^2+44x-48=0$, then the centroid of the triangle whose coordinates are $\left(\alpha ,\frac{1}{\alpha }\right), \left(\beta ,\frac{1}{\beta }\right)$ and $\left(\gamma ,\frac{1}{\gamma }\right)$ is

Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$ be a polynomial whose roots are all negative integer, if $a+b+c+d=2009$, then $d$ is 

If radii of three concentric circles are related as $r_1\left(r_2+r_3\right)+r_2\left(r_3+r_1\right)+r_3\left(r_1+r_2\right)=118$, $r_1{r}_2\left(r_1+r_2\right)+r_2{r}_3\left(r_2+r_3\right)+r_1{r}_3\left(r_1+r_3\right)=616$ and $\Sigma \left(\frac{{r_1}^2+{r_2}^2}{r_1{r}_2}\right)=\frac{44}{5}$, then the area of enclosed region between largest and smallest circle is 

If $\left|x^2-5\left|x\right|+6\right|=k$ has distinct solutions, then $k$ lies in

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to

If $\alpha ,\beta ,\gamma$ are the roots of the equation $4x^3+12x^2-7x+165=0$ and $\alpha +5,\beta +5,\gamma +5$ are the roots of the equation $a{x}^3+b{x}^2+cx+d=0$ then the product of the roots of the second equation is

If $\alpha ,\beta ,\gamma$ are the roots of the equation $3x^3-26x^2+52x-24=0$ such that $\alpha ,\beta ,\gamma$ are in geometric progression and $\alpha <\beta <\gamma$, then $3\alpha +2\beta +\gamma =$

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-5x^2-2x+24=0$ then $\frac{\beta \gamma }{\alpha }+\frac{\gamma \alpha }{\beta }+\frac{\alpha \beta }{\gamma }=$

Let the transformed equation of $2x^4-8x^3+3x^2-1=0$ so that the term containing the cubic power of $x$ is absent be $2x^4+b{x}^2+cx+d=0$. Then $b=$

If $1+\sqrt{2}$ and $2-i$ are the roots of the equation $x^4+b{x}^3+c{x}^2+dx+e=0$ where $b,c,d,e$ are rational numbers, then the roots of the equation $b{x}^2+cx+d=0$ are

If $\frac{5}{2}$ is the sum of two roots of the equation $6x^6-25x^5+31x^4-31x^2+25x-6=0$ then the sum of all non-real roots of the equation is

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and ${\alpha }^3+{\beta }^3+{\gamma }^3=5$, then $r=$

Let $c>0$ and $d<0$. For the equation $x^4+\left(b+c\right)x^3+\left(c+d+bc\right)x^2+\left(c^2+bd\right)x+cd=0$

The cubic polynomial with leading coefficient unity all whose roots are $3$ units less than the roots of the equation $x^3-3x^2-4x+12=0$ is denoted as $f\left(x\right)$, then $f^{\text{'}}\left(x\right)$ is equal to:

The set of values of $k\left(k\in R\right)$ for which the equation $x^2-4\left|x\right|+3-\left|k-1\right|=0$ will have exactly four real roots, is:

Let $S$ denote the set of all real values of '$x$' such that $\left(x^{2010}+1\right)\left(1+x^2+x^4+\cdots \dots +x^{2008}\right)=2010x^{2009}$ then

Solve the equation $18x^3+81x^2+\lambda x+60=0$, one root being half the sum of the other two. Hence find the value of $\lambda$.

${\left(x-\frac{1}{x}\right)}^2+p\left|x+\frac{{\displaystyle 1}}{{\displaystyle x}}\right|+29=0, p\in R$ exactly four distinct real solutions, then the true set of values of $p$ is

The difference of the maximum real root and the minimum real root of the equation ${\left(x^2-5\right)}^4+{\left(x^2-7\right)}^4=16$ is

$x^7-2x+3=P\left(x\right)$. Find number of real roots.