What is with f(x)?

Algebra Level 4

f ( 1 ) = 2 , f ( 2 ) = 4 , f ( 3 ) = 6 , f ( 4 ) = 8 , f ( 5 ) = 10 f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10

Let f f denote a fifth degree monic polynomial that satisfy the equations above. Then what is the value of f ( 6 ) f(6) ?


The answer is 132.

Mayank Patel by mayank patel , 21, India

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2 solutions

Anand O R
Aug 16, 2015

Let g ( x ) = f ( x ) − 2 x g(x) = f(x) - 2x

Clearly , g ( 1 ) = g ( 2 ) = g ( 3 ) = g ( 4 ) = g ( 5 ) = 0 g(1) = g(2) = g(3) = g(4) = g(5) = 0

⇒ \Rightarrow f ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) ( x − 5 ) + 2 x f(x) = (x-1) (x-2) (x-3) (x-4) (x-5) + 2x

Putting x = 6 x=6 ,

f ( 6 ) = f(6) = 132 \boxed { 132 }

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Mayank Patel
Jun 25, 2015

On observing we can see f(x)=2x=(x-1)(x-2)(x-3)(x-4)(x-5)+2x Hence f(6)=132

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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