How do we find the value of k?

Question

anonymous · Answer

$$(3x+k)^3 + (4x-7)^2 $$

anonymous · Answer

has a remainder of 33 when divided by (x-3), how do we find the value of k?

a_clan · Answer

(3x+k)3+(4x−7)2   +  33
----------------------   =  some Integer , say I
          (x-3)

anonymous · Answer

why did you add 33 to the expression?

a_clan · Answer

Because 33 remained when we divided the expression by (x-3). So adding it to the expression will cause the expression to be completely divisible by (x-3).

a_clan · Answer

Sorry, it had to be subtracted.

anonymous · Answer

:o yeah

a_clan · Answer

(3x+k)3+(4x−7)2 - 33
 ---------------------- = some Integer , say I 
             (x-3)

anonymous · Answer

whats $$(3x+k)^3 ?$$

a_clan · Answer

Expanding it is the only way it seems.

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

anonymous · Answer

i've expanded it, but now im lost. how do we find k from here?

anonymous · Answer

a_clan, do u know how to solve this using the remainder theorem? :)

a_clan · Answer

f(x)=(3x+k)^3+(4x−7)^2
According to remainder theorem,
f(3) = 33  
(9+k)^3 + (12 - 7)^2 = 33
  

Rest is simple calculation.

anonymous · Answer

I got f(x) = 721 + k^3
is it incorrect?

a_clan · Answer

k^3 + 27k^2 + 243k + 721 = 0