Cubic Polynomial

Q.

Let be $\alpha$ root of the equation $x^2+x+1=0$ and the matrix $A=$ $\begin{array}{l}\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& {\alpha }^4\end{array}\right]\end{array}$ then the matrix $A^{31}$ is equal to

1. $A$

2. $A^2$

3. $A^3$

4. $I_3$

Q. The equation $3x^2-12x+(n-5)=0$ has equal roots. Find the value of n.

Q.

If $x^3+8xy+y^3=64$, then $\frac{dy}{dx}=$

1. $\frac{-(3x^2+8y)}{(8x+3y^2)}$

2. $\frac{(3x^2+8y)}{(8x+3y^2)}$

3. $\frac{(3x+8y)}{(8x^2+3y)}$

4. None of these

Q. The number of distinct real roots of the cubic polynomial equation $x^3-3x^2+3x-1=0$ is

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

Q. If one of the roots of the quadratic equation $x^2-x-1=0$ is $\alpha$, then its other root is

1. ${\alpha }^2-3\alpha$
2. ${\alpha }^3-2\alpha$
3. ${\alpha }^2-2\alpha$
4. ${\alpha }^3-3\alpha$

Q.

If $\alpha$and $\beta$ are two real roots of the equation $x^3-p{x}^2+qx+r=0$ satisfying the relation $\alpha \beta +1=0$, then find the value of $r^2+pr+q$.

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Q. If $x_1,x_2,x_3$ be the roots of the equation $x^3-x+1=0$, then the value of $\left(\frac{1+x_1}{1-x_1}\right)\left(\frac{1+x_2}{1-x_2}\right)+\left(\frac{1+x_1}{1-x_1}\right)\left(\frac{1+x_3}{1-x_3}\right)+\left(\frac{1+x_2}{1-x_2}\right)\left(\frac{1+x_3}{1-x_3}\right)$ is

Q. The roots of the equation $x^3+8x^2+19x+12=0$ are

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

Q. The number of distinct real zeros of the cubic polynomial $f\left(x\right)=x^3-x^2+x-1$ is

Q. The number of points of intersection of $f\left(x\right)=x^3-9x^2+27x-27$ and $g\left(x\right)=x$ is

Q. If $x^2+px+1$ is a factor of $2co{s}^2\theta x^3+2x+sin2\theta$, then

1. $\theta =n\pi +\frac{\pi }{2},n\in I$
2. $\theta =\frac{n\pi }{2},n\in I$
3. $\theta =n\pi ,n\in I$
4. $\theta =2n\pi ,n\in I$

Q. If one of the roots of the equation $-a{x}^3+b{x}^2+cx+d=0\forall a,b,c,d\in R^{+}$is negative, then the number of positive roots is and the number of imaginary roots is

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

Q. The number of distinct zeros of the cubic polynomial $f\left(x\right)=x^3-9x^2+26x-24$ is

Q. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having $\sqrt[3]{37}+\sqrt[3]{349}$ as one of its roots. Then the product of all the roots of p(x) = 0 is

1. 49
2. 63
3. 7
4. 56

Q. Which among the following is a cubic polynomial?

1. $x^3-3x^{-1}+5$
2. $x^4+3x^3+4x^2-5x+4$
3. $2x^3+5x^2+6x+1$
4. $x^2+x+1$

Q. The cubic polynomial $f\left(x\right)=x^3-5x^2+8x-4$has distinct zeros.

Q. The cubic polynomial $f\left(x\right)=x^3-8x^2+19x-12$ has distinct zeros.

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

Q. Total number of roots for any cubic polynomial equation is always equal to $3$.

1. False
2. True

Q. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having $\sqrt[3]{37}+\sqrt[3]{349}$ as one of its roots. Then the product of all the roots of p(x) = 0 is

1. 49
2. 7
3. 63
4. 56

Q. If one of the roots of the equation $a{x}^3-b{x}^2+cx+d=0\forall a,b,c,d\in R^{+}$ is positive, then the number of negative roots is and the number of imaginary roots is

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

Q. The set of zeros of the cubic polynomial $f\left(x\right)=x^3-2x^2+x$ is

1. $\{0,1,3\}$
2. $\{0,1,2\}$
3. $\{0,1\}$

Q. If $x^3+5x^2+px+q=0$ and $x^3+7x^2+px+r=0$ have two roots in common and their third roots are ${\gamma }_1$ and ${\gamma }_2$ respectively, then the value of $|{\gamma }_1+{\gamma }_2|$ is

1. $13$
2. $12$
3. $17$
4. $5$

Q. The number of distinct real roots of the cubic polynomial equation $x^3-3x^2+3x-1=0$ is

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

Q. The number of distinct real roots of the cubic polynomial equation $x^3-3x^2+3x-1=0$ is

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

Q.

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x^2-1=0$, then the value of $\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }$ is equal to:

1. $-7$

2. $-2$

3. $-6$

4. $-5$

Q.

Sum of the real roots in the equation $x^2|x|-17x^2+95|x|-175=0$ is:

1. None

2. $34$

3. $0$

4. $17$

Q. If $x^3-2x^2-5x+6=0$ and $a,b,c$ are its roots, then match the following with their respective answers.

1. $8$
2. $14$
3. $20$

Q.

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3+2x^2+3x+1=0$. Find the constant term of the equation whose roots are $\frac{1}{{\beta }^3}+\frac{1}{{\gamma }^3}-\frac{1}{{\alpha }^3},\frac{1}{{\gamma }^3}+\frac{1}{{\alpha }^3}-\frac{1}{{\beta }^3},\frac{1}{{\alpha }^3}+\frac{1}{{\beta }^3}-\frac{1}{{\gamma }^3}$.

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Q. If the zeroes of monic cubic polynomial are $3,5$ and $6,$ then the cubic polynomial is

1. $3x^3-42x^2+189x-270$
2. $2x^3-14x^2+60x-78$
3. $2x^3-28x^2+126x-180$
4. $x^3-14x^2+63x-90$

Q.

Find all the real zeros of the polynomial. Use the quadratic formula if necessary.

$P\left(x\right)=5x^3+7x^2-4x-6$