Polyhow
Denote as a fourth degree monic polynomial such that . Calculate .
The answer is 24.
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2 solutions
Moderator note:
Nice standard approach.
Bonus question : Can you generalize this?
Since the Polynomial is a fourth degree Polynomial, it will have 4 roots.
It is given that 1,2 and 3 are the zeroes of the polynomial. Let the fourth root be a
p ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − a )
Putting x to be 0, We get
p ( 0 ) = ( 0 − 1 ) ( 0 − 2 ) ( 0 − 3 ) ( 0 − a ) = 6 a
Now putting x to be 4, We get
p ( 4 ) = ( 4 − 1 ) ( 4 − 2 ) ( 4 − 3 ) ( 4 − 1 ) = 2 4 − 6 a
Adding Both, We get 2 4 − 6 a + 6 a = 2 4
Answer:- 24. QED.
Cheers!