R. D. Sharma Solutions for Chapter: Polynomials, Exercise 1: EXERCISE 2.1

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, then evaluate: $\alpha -\beta$.

If $\alpha -\beta =k$, then find $\frac{\sqrt{b^2-4ac}}{k}$ ?.

If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the zeroes of the quadratic polynomial $\mathrm{f}\left(x\right)=a{x}^2+bx+c$, and $\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}=\frac{-\sqrt{{\mathrm{b}}^2-4\mathrm{kc}}}{\mathrm{c}}$, find $\mathrm{k}$?

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, and $\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta =-\left(\frac{b}{c}+\frac{2k}{a}\right)$, then find $k ?$

$\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$. If ${\alpha }^2\beta +\alpha {\beta }^2=\frac{k}{a^2}$, then, find the value of $k$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, if $\frac{{\left(b^2-2ac\right)}^2-a^4\left({\alpha }^4+{\beta }^4\right)}{a^2{c}^2}=k$, find the value of $k$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, then evaluate $\frac{1}{a\alpha +b}+\frac{1}{a\beta +b}$.

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, then evaluate: $\frac{\beta }{a\alpha +b}+\frac{\alpha }{a\beta +b}$

If $\frac{\beta }{a\alpha +b}+\frac{\alpha }{a\beta +b}=k$, then what is the value of $\left(\frac{-a}{2}\right)k$

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, then evaluate $a\left(\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }\right)+b\left(\frac{\alpha }{\beta }+\frac{\beta }{\alpha }\right)$.