A Question involving cubic functions. Not!

Algebra Level 3

The remainder when f ( x ) f(x) is divided by x x is 12. Given that f ( x + 1 ) − f ( x − 1 ) = 12 x 2 − 12 x − 42 f(x+1) - f(x-1) = 12x^{2} -12x - 42 , find the remainder when f ( x ) f(x) is divided by x − 2 x - 2 .

Source: IMOYA 2017


The answer is -30.

Giwon Kim by Giwon Kim , 21, Philippines

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1 solution

Giwon Kim
Sep 16, 2017

Let x = 1 x = 1 at f ( x + 1 ) − f ( x − 1 ) = 12 x 2 − 12 x − 42 f(x+1)-f(x-1)=12{ x }^{ 2 }-12x-42 ,

Remainder theorem lets you know that

f ( x ) x = f ( 0 ) = 12 \frac { f(x) }{ x }= f(0) = 12

also,

f ( x ) x − 2 = f ( 2 ) \frac { f(x) }{ x-2 } = f(2) .

then f ( 2 ) = − 30 \boxed{f(2) = -30 } .

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