Let f(x)= 2x^n + a be a polynomial such that f(2) = 26 and f(4)=138 then f(-2)=?

Answer :

We have f ( x ) =  2xⁿ  +  a 

And f ( 2 ) = 26  , f ( 4 ) = 138

Now we substitute x  =  2 , in given equation , we get

f ( 2 )  =  2 $\times$  2ⁿ  +  a 

2 $\times$  2ⁿ  +  a   =  26                             ----------- (  1 )                         ( As given f (  2 ) =  26 )

And

Now we substitute x  = 4 , in given equation , we get

f ( 4 )  =  2 $\times$  4ⁿ  +  a 

2 $\times$  4ⁿ  +  a   =  138                            ----------- (  2 )                         ( As given f (  4 ) =  138 )

Now we subtract equation 1 from equation 2 , we get

2 $\times$  4ⁿ  - 2 $\times$  2ⁿ    =  112

2 (  4ⁿ  - 2ⁿ ) =  112

4ⁿ  - 2ⁿ   =  56                                                 ------------------ (  3 )

Now we check for n  =  2 , we get

4² -  2²  =  16 - 4 =  12
And
Now we check for n  =  3 , we get

4³ -  2³  =  64 - 8 =  56  , So n  =  3 satisfied our equation (  3 )

So,

n  =  3  , Substitute in equation 1 , we get

2 $\times$  2³  +  a   =  26

2 $\times$  8 +  a   =  26

a  + 16  =  26

a  =  10

Now we substitute value of n and a in given equation , we get

f ( x ) =  2x³  + 10  

Now we substitute x  = -2 , in given equation , we get

f ( -2 )  =  2 $\times$( - 2 )³  +  10    =  2 ( - 8 ) +  10   =   -16  + 10  =   -6 

So,

f ( - 2 ) =   - 6                                                                                                                        ( Ans )