Descarte's Rule for Positive Roots

Q.

If the cube roots of unity are $1,\omega ,{\omega }^2$ then the roots of the equation $(x-1{)}^3+8=0$ are

1. $-1,-1+2\omega ,-1-2{\omega }^2$

2. $-1,1-2\omega ,1-2{\omega }^2$

3. $-1,-1,-1$

4. None of these

Q.

If the sum of two roots of $x^3+p{x}^2-qx+r=0$ is zero, then $pq$ is equal to

1. $–r$

2. $r$

3. $2r$

4. $–2r$

Q.

How do you find all roots?

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

Q. Select the correct statements for the equation $x^5+4x^4-3x^2+x-6=0.$

1. The number of negative roots can be $0$
2. The number of negative roots can be $2$
3. The number of negative roots can be $1$
4. The number of negative roots can be $3$

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. Possibility no. 1
2. Possibility no. 2
3. Both of the above

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

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

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

Q. Using Descartes Rule of Signs, the maximum possible no. of real roots for $f\left(x\right)=x^5-6x^2-4x+5$ is:

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

Q. Which of the following statement is true about the equation $x^4+2x^2-8x+3=0?$

1. There are $2$ negative real roots
2. There are $2$ positive real roots, $2$ imaginary roots
3. There are $1$ positive real root, $1$ negative real root, $2$ imaginary roots
4. There are $4$ imaginary roots

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

Q. Which of the following statement is true about the equation $x^4+2x^2-8x+3=0?$

1. There are $2$ positive real roots, $2$ imaginary roots
2. There are $4$ imaginary roots
3. There are $2$ negative real roots
4. There are $1$ positive real root, $1$ negative real root, $2$ imaginary roots

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. Total number of positive terms in f(x)

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

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

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

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

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

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

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

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

Q. Select the correct statements for the equation $x^5+4x^4-3x^2+x-6=0$ ?

1. The number of positive roots can be $3$
2. The number of positive roots can be $4$
3. The number of positive roots can be $1$
4. The number of positive roots can be $2$

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

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

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. None of these
2. $a<0$
3. $a=0$
4. $a>0$

Q. Select the correct statements for the equation $x^5+4x^4-3x^2+x-6=0.$

1. The number of negative roots can be $1$
2. The number of negative roots can be $3$
3. The number of negative roots can be $0$
4. The number of negative roots can be $2$

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

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. The maximum possible number of real roots of the equation $x^5-6x^2-4x+5=0$ is

Q.

Determine the number of positive real roots and imaginary roots for equation 3 $x^7$ + 2 $x^5$ + 4 $x^3$ + 11x - 12 = 0.

1. 4, 3

2. 2, 5

3. 3, 4

4. 1, 6