Descarte's Rule for Positive Roots

Q.

Find the number of positive real roots for equation $3x^5 - 4x^4 - 42x^3 + 56x^2 + 27x - 36 = 0$. If all the roots of above given equation are real.

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Q.

The number of positive real roots for polynomial equation f(x) is given by________.

1. Degree of the equation

2. No. of times sign is changing in f(-x)

3. No. of times sign is changing in f(x)

4. Total number of positive terms in f(x)

Q. Consider the cubic equation $x^3-4x^2+x+6=0$
Statement $1:$ Equation has $2$ Positive real roots, $1$ Negative real root
Statement $2:$ Equation has $1$ Negative real root, $2$ Imaginary roots.

1. Only Statement $1$ is possible.
2. Both Statements are possible.
3. Only Statement $2$ is possible.
4. Both Statements are "NOT" possible.

Q. The mximum number of real roots of the equation $f\left(x\right)=x^6+8x^2-14x+1$ is

1. $0$
2. $1$
3. $6$
4. $2$

Q. Let $f\left(x\right)$ be a polynomial of degree $3$ such that $f\left(k\right)=-\frac{2}{k}$ for $k=2,3,4,5$. Then the value of $52-10f\left(10\right)$ is equal to

Q. By Descarte's rule of sign, the maximum number of positive real roots of equation $x^5-6x^4+7x^3+8x^2+9x+10=0$

Q. Using Descartes' Rule of Signs, the maximum possible number of real roots for $f\left(x\right)=x^5-6x^2-4x+5$ is:

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

Q. By Descarte's rule of sign, the maximum number of negative real roots of equation $x^5-6x^4+7x^3+8x^2+9x+10=0$

Q. Using Descartes Rule of Signs, the maximum possible no. of real roots for $f\left(x\right)=x^3-8x^2-9x+12$ is:

1. No. of negative real roots can be $2$
2. No. of real roots can be $3$
3. No. of positive real roots can be $2$
4. No. of positive real roots can be $1$

Q.

One root of the following given equation $2x^5-14x^4+31x^3-64x^2+19x+130=0$ is

1. 7

2. 5

3. 3

4. 1

Q. If $4x^6+a{x}^3+5x-7=0$ has a maximum of $4$ possible real roots, then which of the following is true?

1. $a<0$
2. $a=0$
3. None of the above
4. $a>0$

Q. Using Descartes Rule of Signs, the maximum possible no. of real roots for $f\left(x\right)=x^6+8x^2-14x+1$ is

Q. Two sets A and B are given as
$A=\{x|x\text{is an integer root of the equation}{x}^5-6x^4+11x^3-6x^2=0\}$
$B=\{x|x\text{is a real root of the equation}a{x}^5+2a{x}^3+2b{x}^2+b=0,a,b\in R\text{such that the given equation have}\text{maximum number of real roots.}\}$
Then, which of the following is correct?

1. $n\left(A\right)<n\left(B\right)$
2. cannot comment because $n\left(A\right)$ varries as the values of $a$ and $b$ vary
3. $n\left(A\right)>n\left(B\right)$
4. $n\left(A\right)=n\left(B\right)$

Q. The maximum possible number of real roots for $f\left(x\right)=x^3-8x^2-9x+12$ is:

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

Q. If for the cubic equation $x^3-4x^2+x+6=0$, the possible no. of positive real roots are $2\text{or}0$ and the possible no. of negative real roots is $1$, then which of the following can be true?

1. Number of positive roots$:0$
   Number of imaginary roots:$3$
2. Number of positive roots$:0$
   Number of imaginary roots$:2$
3. Number of positive roots$:2$
   Number of imaginary roots$:0$
4. Number of positive roots$:2$
   Number of imaginary roots$:3$