Recent questions tagged roots

If $\alpha, \beta, \gamma$ are roots of $x^3 + ax + 1 = 0$, then$ \frac{\alpha }{\beta }+\frac{\beta }{\alpha }+\frac{\beta }{\gamma }+\frac{\gamma }{\beta }+\frac{\alpha...

For a polynomial of the form $a_{0}+a_{1} x+\cdots+a_{m} x^{m}$ having degree $m>1$, it is known that the coefficients satisfy the relation $\frac{a_{i-1}}{a_{i-1}+a_{i}}...

The roots of the equation $x^{4}+x^{3}+x^{2}+x+1= 0$ arethe vertices of a square.(some of) the vertices of a regular pentagon.(some of) the vertices of a regular hexagon....

If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3}+6 x+1=0$, then prove that $$ \frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\gamma}+\frac{\gamma}{...

Let $c$ be a positive real number for which the equation $$ x^{4}-x^{3}+x^{2}-(c+1) x-\left(c^{2}+c\right)=0 $$ has a real root $\alpha$. Prove that $c=\alpha^{2}-\alpha$...

The number of real roots of the polynomial $x^{3}-2 x+7$ is$0$$1$$2$$3$

Consider the equation $P(x) =x^3+px^2+qx+r=0$ where $p,q$ and $r$ are all real and positive. State which of the following statements is always correct.All roots of $P(x) ...

The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to$0$$1$$3$$4$

Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ i...

Let $x^2-2(4k-1)x+15k^2-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$

If $\alpha$ and $\beta$ be the roots of the equation $x^2+3x+4=0$, then the equation with roots $(\alpha + \beta)^2$ and $(\alpha – \beta)^2$ is$x^2+2x+63=0$$x^2-63x+2=0$...

If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...

If one root of a quadratic equation $ax^2+bx+c=0$ be equal to the $n^{th}$ power of the other, then$(ac)^{\frac{n}{n+1}} +b=0$$(ac)^{\frac{n+1}{n}} +b=0$$(ac^{n})^{\frac{...

Let $p,q,r,s$ be real numbers such that $pr=2(q+s)$. Consider the equations $x^2+px+q=0$ and $x^2+rx+s=0$. Thenat least one of the equations has real rootsboth these equa...

Let $x^{2}-2(4k-1)x+15k^{2}-2k-7>0$ for any real value of $x$. Then the integer value of $k$ is$2$$4$$3$$1$

If $\alpha$ and $\beta$ be the roots of the equation $x^{2}+3x+4=0,$ then the equation with roots $(\alpha+\beta)^{2}$ and $(\alpha-\beta)^{2}$ is$x^{2}+2x+63=0$$x^{2}-63...

If $r$ be the ratio of the roots of the equation $ax^{2}+bx+c=0,$ then $\frac{r}{b}=\frac{r+1}{ac}$$\frac{r+1}{b}=\frac{r}{ac}$$\frac{(r+1)^{2}}{r}=\frac{b^{2}}{ac}$$\lef...

If one root of a quadratic equation $ax^{2}+bx+c=0$ be equal to the n th power of the other, then$(ac)^{\frac{n}{n+1}}+b=0$$(ac)^{\frac{n+1}{n}}+b=0$$(ac^{n})^{\frac{1}{n...

Let $p,q,r,s$ be real numbers such that $pr=2(q+s).$ Consider the equations $x^{2}+px+q=0$ and $x^{2}+rx+s=0.$ Thenat least one of the equations has real roots.both these...

The sum of the squares of the roots of $x^2-(a-2)x-a-1=0$ becomes minimum when $a$ is$0$$1$$2$$5$

Suppose all the roots of the equation $x^3 +bx-2017=0$ (where $b$ is a real number) are real. Prove that exactly one root is positive.