question_answer

For \[{{x}^{2}}+2x+5\]to be a factor of\[{{x}^{4}}+\alpha {{x}^{2}}\text{+}\beta ,\] the values of \[\alpha \] and \[\beta \] should respectively be_____.

A)  2, 5                            

B)  5, 25  

C)         6, 25              

D)         5, 2    

Correct Answer: C

Solution :

For \[{{x}^{2}}+2x+5\] to be a factor of \[{{x}^{4}}+\alpha {{x}^{2}}+\beta ,\] remainder should be zero.   Now, remainder should be equal to zero. \[\therefore \]    \[\alpha -6=0\] and  \[\beta -25=0\] \[\Rightarrow \]            \[\alpha =6\] and \[\beta =25\]