Question

plz solve this problem fast..​

Answer 1

$\large\underline{\sf{Given \:Question - }}$

Identify the zeroes of the given polynomial

$\rm :\longmapsto\:p(z) = {4z}^{2} - 15z\pi \: - 4 {\pi}^{2}$

$\large\underline{\sf{Solution-}}$

Given polynomial is

$\rm :\longmapsto\:p(z) = {4z}^{2} - 15z\pi \: - 4 {\pi}^{2}$

can be rewritten as

$\rm :\longmapsto\:p(z) = {4z}^{2} - 16z\pi \: + z\pi \: - 4 {\pi}^{2}$

$\rm :\longmapsto\:p(z) = 4z(z - 4\pi \:) + \pi \:(z - 4\pi \:)$

$\rm :\longmapsto\:p(z) = (z - 4\pi \:)(4z + \pi \:)$

To find zeroes of p(z), Substitute

$\rm :\longmapsto\:p(z) = 0$

Thus,

$\rm :\longmapsto\:(z - 4\pi \:)(4z + \pi \:) = 0$

$\bf\implies \:z = 4\pi \: \: \: \: or \: \: \: \: z \: = - \: \dfrac{\pi}{4}$

Verification :-

Given polynomial is

$\rm :\longmapsto\:p(z) = {4z}^{2} - 15z\pi \: - 4 {\pi}^{2}$

Case :- 1

$\red{ \boxed{ \sf{ \:z = 4\pi \:}}}$

On substituting the value of z, in above polynomial, we get

$\rm :\longmapsto\:p(4\pi \:) = {4(4\pi \:)}^{2} - 15(4\pi \:)\pi \: - 4 {\pi}^{2}$

$\rm :\longmapsto\:p(4\pi \:) = {64\pi \:}^{2} - {60\pi}^{2} \: - 4 {\pi}^{2}$

$\rm :\longmapsto\:p(4\pi \:) = {64\pi \:}^{2} - {64\pi}^{2} \:$

$\rm :\longmapsto\:p(4\pi \:) = 0$

Hence, Verified

Case :- 2

$\red{ \boxed{ \sf{ \:z = \: - \: \frac{\pi}{4 \:} \:}}}$

So, on substituting the value of z, in above polynomial, we get

$\rm :\longmapsto\:p\bigg[ - \dfrac{\pi \:}{4} \bigg]= 4 {\bigg[ - \dfrac{\pi \:}{4} \bigg]}^{2} + 15\bigg[\dfrac{\pi \:}{4} \bigg]\pi \: - 4 {\pi}^{2}$

$\rm :\longmapsto\:p\bigg[ - \dfrac{\pi \:}{4} \bigg]= \dfrac{ {\pi \:}^{2} }{4} + \dfrac{15 {\pi \:}^{2} }{4} - 4 {\pi}^{2}$

$\rm :\longmapsto\:p\bigg[ - \dfrac{\pi \:}{4} \bigg]= \dfrac{ {16\pi \:}^{2} }{4} - 4 {\pi}^{2}$

$\rm :\longmapsto\:p\bigg[ - \dfrac{\pi \:}{4} \bigg]= {4\pi}^{2} - 4 {\pi}^{2}$

$\rm :\longmapsto\:p\bigg[ - \dfrac{\pi \:}{4} \bigg]= 0$

Hence, Verified

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: Hence, \: zeroes \: are-\begin{cases} &\sf{z = 4\pi \:} \\ \\ &\sf{z \: = - \: \dfrac{\pi \:}{4} } \end{cases}\end{gathered}\end{gathered}$