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

In which of the following cases, the given equations has atleast one root in the indicated interval?

Solve: ${\left(1+x\right)}^{\frac{2}{3}}+{\left(1-x\right)}^{\frac{2}{3}}=3{\left(1-x^2\right)}^{\frac{1}{3}}$.

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation $x^4-8x^3+11x^2+32x-60=0$ such that $\alpha <\beta <\gamma <\delta$, then $4\alpha +3\beta +2\gamma +\delta =$

For $x\le 2$, solve $x^3{3}^{\left|x-2\right|}+3^{x+1}=x^3·3^{x-2}+3^{\left|x-2\right|+3}$.

Let $A$ line $2y=x$ intersect curve $2x^{3}y+x^2=4-8x{y}^2$ at $A_1,A_2,A_3\dots .A_n\left(n\in N\right)$ then $\prod _{i=1}O{A}_i$ is equal to (Where $O$ being origin)

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:

If $x=3+3^{1/3}+3^{2/3}$ then $x^3-9x^2+18x+12=$

If $a,b,c$ are the roots of the cubic $x^3-x^2+2x-3=0$, then find the absolute value of the expression $\frac{a}{a+bc}+\frac{b}{b+ac}+\frac{c}{c+ab}$.

If $\alpha$ and $\beta$ are the real roots of the equation $p^{\frac{1}{3}}-\frac{1}{p^{\frac{1}{3}}}-\sqrt{2}=0$, then which of the following statement(s) is(are) correct?

$f\left(x\right)$ be a polynomial of degree at most $7$ which leaves remainders $-1$ and 1 upon division by ${\left(x-1\right)}^4$ and ${\left(x+1\right)}^4$ respectively. If the sum of pairwise product of all roots of $f\left(x\right)=0$ is $n$, then the value of $n$ is

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x-1=0$ then the value of the expression $\sum \frac{{\left(\alpha +\beta \right)}^2}{{\gamma }^3-2}$ equals

For the equation $\left|x^2+{\left(k\right)}^{\frac{1}{3}}\right|=16\left|x\right|, k\in R^{+}$ to have a real solution the greatest value of $k$ is

The number of ordered pairs $\left(x, y\right)$ satisfying the following system of equations $2^{y-x}\left(x+y\right)=1$ and ${\left(x+y\right)}^{x-y}=2$ can be

Consider the equation $\left|x^2-9\right|-\left|x-\lambda \right|=0, \lambda \in R.$

Let $m$ be the number of solutions of the equation when $\lambda =2$ and $n$ be the number of solutions of the equation when $\lambda =3$ and $p$ be the number of solutions of the equation when $\lambda =-10.$ Then the value of $\frac{n^m+m^p}{m^n},$ upto two places of decimal, is

If $a, b, c$ are three distinct non-zero real numbers satisfying the equations $\frac{1}{a}+\frac{1}{a-1}+\frac{1}{a-2}=1, \frac{1}{b-2}+\frac{1}{b}+\frac{1}{b-1}=1$ and $\frac{1}{c-1}+\frac{1}{c-2}+\frac{1}{c}=1.$

Then the value of $\frac{\left(a+b+c\right)+\left(ab+bc+ca\right)}{abc}$ is

Given that $a$ is a root of the equation $x^2-x-3=0$ then $\frac{9\left(a^3+1\right)}{a^5-a^4-a^3+a^2}$ equals

Number of solutions of the equation $\sqrt{3x-2}+\sqrt{4x+4}=\sqrt{5x+1}+\sqrt{4x-5}$ is

Use hit and trial method to solve the below equation.

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

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