ROOTS OF COMPLEX NUMBERS (Excercise Problems)

\[\LARGE{\color {purple} {PART- B}}\]
\[\color {purple} {1.}\ \color {red} {If \ ω\ is\ the\ cube\ roots\ of unity,\ what\ is\ the\ value\ of}\ ω( ω\ +\ 1)\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ Given\ ω\ is\ the\ cube\ roots\ of unity\ \hspace{20cm}\]
\[i.\ e\ 1\ +\ ω\ +\ ω ^2\ =\ 0\ ——\ (1)\ \hspace{10cm}\]
\[ω( ω\ +\ 1)\ =\ ω ^2\ +\ ω\ =\ -\ 1\ (using\ (1))\ \hspace{10cm}\]
\[\LARGE{\color {purple} {PART- C}}\]
\[\color {purple} {2:}\ \color {red} {Solve}\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{18cm}\]
\[\color {blue}{Solution:}\ Given\ x^5 +\ x^3\ +\ x^2\ +\ 1\ =\ 0\ \hspace{15cm}\]
\[x^3(x^2 +\ 1)\ +\ 1( x^2\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[(x^3\ +\ 1)\ ( x^2\ +\ 1)\ =\ 0\ \hspace{10cm}\]
\[\color {green} {Case\ ( i ) :}\ \hspace{16cm}\]
\[x^3\ +1 =\ 0\ \hspace{10cm}\]
\[ x^3\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{3}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{3})\ +\ i\ sin\ (\frac{π\ + 2kπ}{3})\ where\ k\ =\ 0,\ 1,\ 2\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{3}\ +\ i\ sin\ \frac{π}{3}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{3}\ +\ i\ sin\ \frac{3π}{3}\ =\ -1\]
\[When\ k = 2,\ \hspace{2cm}\ x\ =\ cos\ \frac{5π}{3}\ +\ i\ sin\ \frac{5π}{3}\]
\[\color {green} {Case\ ( ii ) :}\ \hspace{16cm}\]
\[x^2\ +1 =\ 0\ \hspace{10cm}\]
\[ x^2\ =\ -\ 1\ \hspace{10cm}\]
\[\ x\ =\ (-\ 1)^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ (cos\ π\ +\ i\ sin\ π)^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ (cos\ (π\ + 2kπ) +\ i\ sin\ (π\ + 2kπ))^\frac{1}{2}\ \hspace{10cm}\]
\[ =\ cos\ (\frac{π\ + 2kπ}{2})\ +\ i\ sin\ (\frac{π\ + 2kπ}{2})\ where\ k\ =\ 0,\ 1\ \hspace{5cm}\]
\[The\ roots\ are\]
\[When\ k = 0,\ \hspace{2cm}\ x\ =\ cos\ \frac{π}{2}\ +\ i\ sin\ \frac{π}{2}\]
\[When\ k = 1,\ \hspace{2cm}\ x\ =\ cos\ \frac{3π}{2}\ +\ i\ sin\ \frac{3π}{2}\]
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