Relation Between Roots and Coefficients for Higher Order Equations

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

1. $\sum {\beta }^2{\gamma }^2=-14$
2. $\sum \frac{1}{\alpha }=-\frac{4}{5}$
3. $\sum {\alpha }^2=1$
4. $\sum \alpha =3$

Q. For the equation $p{x}^4+q{x}^3+r{x}^2+sx+t=0;p>0$, all the roots are positive real numbers, then which of the following is/are true?

1. $t>0$
2. $q<0$
3. $s<0$
4. $r>0$

Q.

If $\alpha ,\beta$are the roots of the equation $x^2-px+q=0,$then the values of $(\alpha +\beta )x-({\alpha }^2+{\beta }^2)x^2/2+({\alpha }^3+{\beta }^3)x^3/3+\dots .$is

Q. If $a_1,a_2,a_3,a_4,a_5$ are the roots of the equation $6x^5-41x^4+97x^3-97x^2+41x-6=0,$ such that $|a_1|\le |a_2|\le |a_3|\le |a_4|\le |a_5|,$ then which of the following is/are correct?

1. $a_1,a_2,a_3$ are in H.P.
2. the equation has three real roots and two imaginary roots.
3. $a_3,a_4,a_5$ are in A.P.
4. $a_1,a_2,a_3$ are in G.P.

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

1. $\sum \alpha =3$
2. $\sum \frac{1}{\alpha }=-\frac{4}{5}$
3. $\sum {\beta }^2{\gamma }^2=-14$
4. $\sum {\alpha }^2=1$

Q. If the product of two roots of the equation $x^4-5x^3+5x^2+5x-6=0$ is $3$, then which of the following is/are correct?

1. The product of all positive roots will be $6.$
2. The equation has three negative roots.
3. The equation has only one negative root.
4. The product of all negative roots will be $6.$

Q.

If the sum of the two roots of the equation $4x^3+16x^2-9x-36=0$ is zero, then the roots are

1. 1, 2, -2

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

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

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

Q. Any equation having more roots than it's degree is called an identity.

1. False
2. True

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+lx+m=0$, then the value of ${\alpha }^3+{\beta }^3+{\gamma }^3$ is

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

Q.

If one root of the cubic equation $x^3-30x+133=0$ is $\frac{7+3\sqrt{3}i}{2}$. Find the real root of the cubic equation.

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Q. Let $p,q,r$ be roots of cubic equation $x^3+2x^2+3x+3=0$, then

1. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=44$
2. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=6$
3. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=38$
4. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=5$

Q. If $b^2<2ac$ and $a,b,c,d\in R,$ then the number of real roots of the equation $a{x}^3+b{x}^2+cx+d=0$ are

Q. For an equation $x^3-x^2-6x+6=0$, what is the value of $\sum a^2$, if $a,b,c$ are its roots ?

1. $13$
2. $14$
3. $15$
4. $16$

Q. If all the roots of the equation $x^3+px+q=0,p,q\in R,q\ne 0$ are real, then which the following is correct?

1. $p<0$
2. $p>0$
3. $p=0$
4. $p\le 0$

Q. For the equation, $x^7+3x^6-2x^5+4x^4-4x^3+3x^2-2x+1=0$
, the value of the sum of the roots taken $3$ at a time is:

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

Q. If all the roots of the equation $x^3+px+q=0,p,q\in R,q\ne 0$ are real, then which the following is correct?

1. $p>0$
2. $p<0$
3. $p=0$
4. $p\le 0$

Q. If $\alpha ,\beta and\gamma$ are the real roots of the equation $x^3+5x^2+9x-6=0$, then the value of ${\alpha }^2+{\beta }^2+{\gamma }^2$ is

Q. If the sum of two roots of $x^4-2x^3+4x^2+6x-21=0$ is zero, then which of the following is/are true?

1. the equation has only two real roots
2. one of the roots of the equation is $1+i\sqrt{6}$
3. all roots of the equation are real
4. sum of all the real roots of the equation is $0$

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3$ + 3$x^2$ + 5x - 6 = 0, find the value of $(\alpha -\frac{1}{\beta \gamma })(\beta -\frac{1}{\gamma \alpha })(\gamma -\frac{1}{\alpha \beta }){(\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma })}^{-1}$

1. $\frac{36}{25}$

2. $\frac{25}{6}$

3. $\frac{125}{36}$

4. $\frac{36}{125}$

Q.

If roots of the equation $a{x}^3+b{x}^2+cx+d=0$ remain unchanged by increasing each coefficient by one unit, then

1. a = b = c = d $\ne$ 0

2. $a\ne 0,b+c=0,d\ne 0$

3. $a\ne b\ne c\ne d\ne 0$

4. a = b $\ne$ c = d $\ne$ 0

Q. For the equation, $7x^6-4x^5+3x^4-2x+9=0$, if the sum of the roots taken two at a time is represented by $S_2,$ then the value of $S_2$ is:

1. $\frac{3}{7}$
2. $\frac{-2}{7}$
3. $\frac{-4}{7}$
4. $\frac{4}{7}$

Q. Find the product of all the roots for the equation $3x^3+5x^2+9x=0$.
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Q. If $\alpha ,\beta ,\gamma$ are roots of equation $x^3-x-1=0$, then the equation whose roots are $\frac{1}{\beta +\gamma },\frac{1}{\gamma +\alpha },\frac{1}{\alpha +\beta }$ is -

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

Q. If $\alpha ,\beta ,\gamma ,\delta \in R$ satisfy $\frac{(\alpha +1{)}^2+(\beta +1{)}^2+(\gamma +1{)}^2+(\delta +1{)}^2}{\alpha +\beta +\gamma +\delta }=4$.
If the equation $a_0{x}^4+a_1{x}^3+a_2{x}^2+a_{3}x+a_4=0$ has the roots $(\alpha +\frac{1}{\beta }-1),(\beta +\frac{1}{\gamma }-1),(\gamma +\frac{1}{\delta }-1),(\delta +\frac{1}{\alpha }-1)$, then the value of $\frac{a_2}{a_0}$ is

Q. Let $r,s,t$ and $u$ be the roots of the equation $x^4+A{x}^3+B{x}^2+Cx+D=0;$
$A,B,C,D\in R.$ If $rs=tu$, then $A^{2}D$ is equal to

1. $B^2$
2. $C^2$
3. $0$
4. $BC$

Q. Which of the following is/are the roots of the equation, $3x^4+25x^3+6x^2+25x+3=0$ ?

1. $\frac{-25-\sqrt{589}}{6}$
2. $\frac{-25+\sqrt{589}}{6}$
3. $0$
4. All of the above

Q. Let $r,s,t$ and $u$ be the roots of the equation $x^4+A{x}^3+B{x}^2+Cx+D=0;$
$A,B,C,D\in R.$ If $rs=tu$, then $A^{2}D$ is equal to

1. $BC$
2. $B^2$
3. $0$
4. $C^2$

Q. Any equation having more roots than it's degree is called an identity.

1. False
2. True

Q. Find the product of all the roots for the equation $3x^3+5x^2+9x=0$.
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Q. The number of the distinct zeros of the polynomial $f\left(x\right)=x(x-4{)}^3(x-3{)}^2(x-1)$ is