Biquadratic Equations of the form: ax^4+bx^3+cx^2+bx+a=0

Q.

Solve for $x$:

$\frac{\left(x-1\right)}{\left(x-2\right)}+\frac{\left(x-3\right)}{\left(x-4\right)}=\frac{10}{3}$

Q.

Solve for $x$: $\frac{(x-2)}{(x-3)}+\frac{(x-4)}{\left(x-5\right)}=\frac{10}{3},x\ne 3,5$

Q. The roots of the equation, $6x^4-25x^3+12x^2+25x+6=0$ are

1. $\frac{3}{2},-2,-\frac{2}{3},3$
2. $\frac{1}{2},-2,\frac{1}{3},-3$
3. $-\frac{1}{2},2,-\frac{1}{3},3$
4. $\frac{3}{2},2,\frac{2}{3},3$

Q. The number of real roots of the equation $2x^4-4x^3-4x+2=0$ is

1. $2$
2. $4$
3. $0$

Q. Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

1. $(-\frac{1}{2},\frac{1}{2})$
2. $(-\frac{5}{2},\frac{5}{2})$
3. $(-1,1)$
4. $(-3,3)$

Q. The product of all positive real values of $x$ satisfying the equation $x^{\left(16\right({\log }_{5}x{)}^3-68{\log }_{5}x)}=5^{-16}$ is

Q. The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has only $2$ real roots which are negative is

1. $\varphi$
2. $(-\frac{1}{2},\frac{5}{2}]$
3. $[\frac{5}{2},\infty )$
4. $(-\infty ,-\frac{1}{2}]$

Q. A polynomial equation with a degree $4$ will have roots.

1. $0$
2. $4$
3. $0$

Q. The number of real roots of the equation $e^{4x}+{2e}^{3x}-e^x-6=0$ is

1. $0$
2. $2$
3. $1$
4. $4$

Q. The number of real roots of the equation $2x^4-4x^3-4x+2=0$ is

1. $4$
2. $2$
3. $0$

Q.

Solve the equation $x^4-16x^3+86x^2-176x+105=0$. If two roots being $1$ and $7$, Find the sum of the square of other two roots.

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Q. If the equation $x^4 - (k-1)x^2+(2 - k) = 0$ has three distinct real roots, then the possible value(s) of $k$ is/are

Q. The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has $2$ positive and $2$ negative roots is

1. $\varphi$
2. $(\frac{5}{2},\infty )$
3. $(-\frac{1}{2},\frac{5}{2})$
4. $(-\infty ,-\frac{1}{2})$

Q. A polynomial equation with a degree $4$ will have roots.

1. $4$
2. $2$
3. $0$

Q. The roots of the equation $2x^4+x^3-11x^2+x+2=0$ is/are

1. $\frac{-5+\sqrt{3}}{2}$
2. $\frac{1}{4}$
3. $\frac{-3-\sqrt{5}}{2}$
4. $\frac{1}{2}$

Q. If $f\left(x\right)=4x^4+12x^3+c{x}^2+6x+d$ is a perfect square, then

1. Minimum value of $f\left(x\right)$ occurs at $x=-\frac{1}{2}$
2. $d=1$
3. $c=5$
4. $c=13$

Q. The product of all roots of the equation $(x^2+7x+3{)}^2-(x-6\left)\right(x-1\left)\right(x-9)=5$ is

Q. Find the values of m for which equation $3x^2+mx+2=0$ has equal roots. Also, find the roots of the given equation.

Q. The number of real roots of the polynomial equation $x^4-x^2+2x-1=0$ is

Q. Among the given polynomials equations, select the biquadratic polynomial equation(s).

1. $(x^2-4)(2x-x^2)=0$
2. $x^2-x=0$
3. $(x-2)(2x-4)(x-4)(x+1)=0$
4. $-4x^4-x^3+2x-9=0$

Q. Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

1. $(-\frac{5}{2},\frac{5}{2})$
2. $(-1,1)$
3. $(-3,3)$
4. $(-\frac{1}{2},\frac{1}{2})$

Q. If the equation $x^4-(k-1)x^2+(2-k)=0$ has three distinct real roots, then the possible value(s) of $k$ is/are

1. $\{2\sqrt{2},\sqrt{3}-\sqrt{2}\}$
2. $\{\sqrt{5}-1\}$
3. $\{\sqrt{2}-1,2\}$
4. $\left\{2\right\}$

Q. The roots of the equation, $6x^4-25x^3+12x^2+25x+6=0$ are

1. $\frac{1}{2},-2,\frac{1}{3},-3$
2. $-\frac{1}{2},2,-\frac{1}{3},3$
3. $\frac{3}{2},-2,-\frac{2}{3},3$
4. $\frac{3}{2},2,\frac{2}{3},3$

Q. For the equation
$2x^4-3x^3-x^2-3x+2=0$,
Which among the following statement(s) is/are correct?

1. Sum of the real roots = $\frac{5}{2}$
2. Product of real roots = $1$.
3. All roots of the equation are real
4. Exactly two roots of the equation are real

Q. The number of real roots of the equation $e^{4x}+{2e}^{3x}-e^x-6=0$ is

1. $1$
2. $4$
3. $2$
4. $0$

Q. If $f\left(x\right)=4x^4+12x^3+c{x}^2+6x+d$ is a perfect square, then

1. Minimum value of $f\left(x\right)$ occurs at $x=-\frac{1}{2}$
2. $c=13$
3. $d=1$
4. $c=5$

Q. The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has only $2$ real roots which are negative is

1. $\varphi$
2. $[\frac{5}{2},\infty )$
3. $(-\infty ,-\frac{1}{2}]$
4. $(-\frac{1}{2},\frac{5}{2}]$

Q. The product of all roots of the equation $(x^2+7x+3{)}^2-(x-6\left)\right(x-1\left)\right(x-9)=5$ is

Q. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

1. $3$
2. $1$
3. $2$
4. $4$

Q. If $x=\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a root of $x^4+a{x}^3+b{x}^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$ ?