Fundamental Theorem of Algebra

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

1. 3
2. 5
3. 0
4. 4

Q. Find the total possible number of roots of the equation $4x^7+3x^6+2x^5+9x^4+4x^3+x^2+9x+9=0$

1. $7$
2. $0$
3. $3$
4. $4$

Q.

Let $a,b,c$ be real numbers $a\ne$ 0. If $\alpha$ is a root of $a^2{x}^2+bx+c=0,\beta$ is a root of $a^2{x}^2-bx-c=0$ and $0<\alpha <\beta$, then the equation $a^2{x}^2+2bx+2c=0$ has a root $\gamma$ that always satisfies:

1. $\gamma =\alpha +\frac{\beta }{2}$

2. $\gamma =\frac{\alpha +\beta }{2}$

3. $\gamma =\alpha$

4. $\alpha <\gamma <\beta$

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

1. -5
2. 3
3. 6
4. -11

Q.

Sum of the real values of ${}^{\prime }{a}^{\prime }$ for which the equation $(a^2-3a+2)x^2+(a^2-4a+3)x+(a^2-6a+5)=0$ has three distinct roots

1. $11$

2. $1$

3. $10$

4. None

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

1. 0
2. 3
3. 4
4. 5

Q.

What is the degree of the polynomial $f\left(x\right)=x^2-5x+6$ ?

__

Q. Fundamental theorem of algebra states that a polynomial equation of $n$ degree have exactly $n$ roots, either real or imaginary.

1. True
2. False

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

1. 6
2. -5
3. -11
4. 3

Q. The total possible no. of roots of the equation $4x^7+3x^6+2x^5+9x^4+4x^3+x^2+9x+9=0$ is

Q.

The highest index of the variable x occurring in the polynomial $f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+........a_{n-1}x+a_n$​ is called ___________.

1. Order of the polynomial

2. Both of these

3. degree of the polynomial

4. None of these

Q. If the equation $x^3-8x^2+cx+d=0\forall c,d\in R$ has one complex root and one positive root, then select the correct statement.

1. $c,d>0$
2. $c<0,d>0$
3. $c<0,d<0$
4. $c>0,d<0$

Q. The total possible no. of roots of the equation $4x^7+3x^6+2x^5+9x^4+4x^3+x^2+9x+9=0$ is

Q. Find the total possible number of roots of the equation $4x^7+3x^6+2x^5+9x^4+4x^3+x^2+9x+9=0$

1. $4$
2. $3$
3. $0$
4. $7$

Q. If the roots of the equation $(a-1)(x^2+x+1{)}^2=(a+1\left)\right(x^4+x^2+1)$ are real and distinct, then $a$ belongs to

1. $(-\infty ,-2)\cup (2,\infty )$
2. $[-2,2]$
3. $(-\infty ,3]$
4. $\varphi$

Q. The degree of the polynomial function $f\left(x\right)=(x-1{)}^2(x+2{)}^2$ is $4.$

1. True
2. False

Q.

Sum of the real values of ${}^{\prime }{a}^{\prime }$ for which the equation $(a^2-3a+2)x^2+(a^2-4a+3)x+(a^2-6a+5)=0$ has three distinct roots

1. $11$

2. $1$

3. $10$

4. None