JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Higher Degree Equations

The equation \[p(x)={{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+.....+{{a}_{n-1}}x+{{a}_{n}}=0\]…..(i)

Where the coefficients \[{{a}_{0}},{{a}_{1}},.......,{{a}_{n}}\in R\] (or C) and \[{{a}_{0}}\ne 0\] is called an equation of \[{{n}^{th}}\] degree, which has exactly \[n\] roots \[{{\alpha }_{1}},\,{{\alpha }_{2}},........,{{\alpha }_{n}}\in C\], then we can write \[p(x)={{a}_{0}}(x-{{\alpha }_{1}})\,(x-{{\alpha }_{2}})......(x-{{\alpha }_{n}})\]

= \[{{a}_{0}}\{{{x}^{n}}-(\Sigma {{\alpha }_{1}}){{x}^{n-1}}+(\Sigma {{\alpha }_{1}}{{\alpha }_{2}}){{x}^{n-2}}-.....+{{(-1)}^{n}}{{\alpha }_{1}}{{\alpha }_{2}}.....{{\alpha }_{n}}\}\]                                                                                       …(ii)

Comparing (i) and (ii), \[\Sigma {{\alpha }_{1}}={{\alpha }_{1}}+{{\alpha }_{2}}+.......+{{\alpha }_{n}}=-\frac{{{a}_{1}}}{{{a}_{0}}}\]

\[\Sigma {{\alpha }_{1}}{{\alpha }_{2}}={{\alpha }_{1}}{{\alpha }_{2}}+.....+{{\alpha }_{n-1}}{{\alpha }_{n}}=\frac{{{a}_{2}}}{{{a}_{0}}}\]

and so on and \[{{\alpha }_{1}}{{\alpha }_{2}}.....{{\alpha }_{n}}={{(-1)}^{n}}\frac{{{a}_{n}}}{{{a}_{0}}}\]

Cubic equation : When \[n=3\], the equation is a cubic of the form \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\], and we have in this case \[\alpha +\beta +\gamma =-\frac{b}{a};\] \[\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a};\,\text{  }\alpha \beta \gamma =-\frac{d}{a}\]

Biquadratic equation : If \[\alpha ,\beta ,\gamma ,\delta \] are roots of the biquadratic equation \[a{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+e=0\], then \[{{\sigma }_{1}}=\alpha +\beta +\gamma +\delta =-\frac{b}{a}\]

\[{{\sigma }_{2}}=\alpha \beta +\alpha \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta =\frac{c}{a}\]

\[{{\sigma }_{3}}=\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta =-\frac{d}{a}\]

\[{{\sigma }_{4}}=\alpha \beta \gamma \delta =\frac{e}{a}\].

Formation of a polynomial equation from given roots : If \[{{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}},.....{{\alpha }_{n}}\]are the roots of a polynomial equation of degree n, then the equation is

\[{{x}^{n}}-{{\sigma }_{1}}{{x}^{n-1}}+{{\sigma }_{2}}{{x}^{n-2}}-{{\sigma }_{3}}{{x}^{n-3+......+}}{{(-1)}^{n}}{{\sigma }_{n}}=0\]

where \[{{\sigma }_{r}}=\sum{{{\alpha }_{1}}}{{\alpha }_{2}}.....{{\alpha }_{r}}.\]

Cubic equation : If \[\alpha ,\beta ,\gamma \] are the roots of a cubic equation, then the equation is \[{{x}^{3}}-{{\sigma }_{1}}{{x}^{2}}+{{\sigma }_{2}}x-{{\sigma }_{3}}=0\] or \[{{x}^{3}}-(\alpha +\beta +\gamma ){{x}^{2}}+(\alpha \beta +\alpha \gamma +\beta \gamma )x-\alpha \beta \gamma =0\].

Biquadratic Equation : If \[\alpha ,\beta ,\gamma ,\delta \] are the roots of a biquadratic equation, then the equation is

\[{{x}^{4}}-{{\sigma }_{1}}{{x}^{3}}+{{\sigma }_{2}}{{x}^{2}}-{{\sigma }_{3}}x+{{\sigma }_{4}}=0\]

or  \[{{x}^{4}}-(\alpha +\beta +\gamma +\delta ){{x}^{3}}+(\alpha \beta +\alpha \gamma +\alpha \delta \]

\[+\beta \gamma +\beta \delta +\gamma \delta ){{x}^{2}}-(\alpha \beta \gamma +\alpha \beta \delta +\alpha \gamma \delta +\beta \gamma \delta )\]\[x+\alpha \beta \gamma \delta =0\]