Algebraic Equations of Higher Degree

The value of $x$, which satisfies the equation $\left(x-1\right)\left|x^2-4x+3\right|+2x^2+3x-5=0$, is

The sum of all non-integer roots of the equation $x^5-6x^4+11x^3-5x^2-3x+2=0$ is

Let $1,{\zeta }_2 ,{\zeta }_3, . . . , {\zeta }_n$ be the roots of the equation $x^n=1,$ for $n\ge 3$. Then, the sum

$\frac{1}{2-{\zeta }_2}+\frac{1}{2-{\zeta }_3}+...+\frac{1}{2-{\zeta }_n}$ equals to

For the equation $x^2-5\left|x\right|-14=0$,

The roots of the polynomial $x^3-39x^2+471x-1729$ are in an arithmetic progression. Which of the following is a common difference of the progression?

The number of points of intersection of the curves $x^2+8y^2=4$ and $x^2+y^2=1$ is

How many functions $f:N\to N$ satisfy

$lcm\left(f\left(n\right),n\right)-hcf\left(f\left(n\right),n\right)<5?$

Here '$\mathrm{lcm}$' denotes the least common multiple and '$\mathrm{hcf}$' denotes the highest common factor.

Use hit and trial method to solve the below equation.

$3x^3-4x^2+2x+44=0$

Find $X$,if $\mathrm{x}=\sqrt{6+\sqrt{6+\sqrt{6+\dots .\mathrm{ }\mathrm{u}\mathrm{p}\mathrm{ }\mathrm{t}\mathrm{o}\mathrm{ }\mathrm{\infty }}}}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }$

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-7x+7=\mathrm{0 }$, then $\frac{1}{{\alpha }^4}+\frac{1}{{\beta }^4}+\frac{1}{{\gamma }^{\mathrm{4 }}}$ is

The number of points of intersection of $2y=1$ and $y=\sin x$, $-2\pi \le x\le 2\pi$

Suppose $\alpha ,\beta ,\gamma$ are roots of $x^3+x^2+2x+3=0$. If $f\left(x\right)=0$ is a cube polynomial equation whose roots are $\alpha +\beta , \beta +\gamma , \gamma +\alpha$ then $f\left(x\right)=$

The number of real roots of the equation $5+\left|2^x-1\right|=2^x\left(2^x-2\right)$ is : $\mathrm{ }$

The equation $\frac{a^2}{x-a^{\text{'}}}+\frac{b^2}{x-b}+\frac{c^2}{x-c^2}+\dots +\frac{k^2}{x-k^{\text{'}}}=x-m,$ where $a,b,c,\dots ,k,m,a^{\text{'}},b_j,\dots ,k^{\text{'}}$ are all real numbers then 

Find the sum of all the roots of the equation $8x^3-20x^2+6x+9=0$. Given that the equation has multiple roots.

The roots of $x^5-3x^4-5x^3+27x^2-32x+12=0$ are 

If the equation  $x^5-5x^3+5x^2-1=0$ has three equal roots then find this root.

Find the sum of the roots of the equation $8x^4-2x^3-27x^2+6x+9=0$ given that two roots have the same absolute value, but are opposite in sign.

Find the roots of the equation $x^4+x^3-16x^2-4x+48=0$, given that the product of two of the roots is $6$.

Find the integer value of the roots of the equation $6x^3-11x^2+6x-1=0$ given that the roots are in $\mathrm{H}.\mathrm{P}.$