Question

please solve this question step by step

Answer 1

Hola !

∆ By Remainder Theorem : Just assume that
${x}^{3} = {e}^{2} x$
And keep repeatedly solving till you get to the minimum degree of 'x' !

$e {x}^{135} = e( {x}^{27} )^{5} = {e}^{131} {x}^{5} = {e}^{135} x$

Similarly, keep substituting to get :

${e}^{11} {x}^{125} = {e}^{131} {x}^{5} = {e}^{135} x$

${e}^{21} {x}^{115} = {e}^{131} {x}^{5} = {e}^{135} x$


Now, just substitute the values we Concluded in the Remainder Theorem :

$p( {e}^{2} x) = {e}^{135} x + {e}^{135} x - {e}^{135} x - {e}^{135} x + 1$

Hence, by Remainder Theorem, the Remainder is 1