What will be the value of P(12)+P(−8) if P(x)=x^4+ax^3+bx^2+cx+d provided that P(1)=10, P(2)=20, P(3)=30?

Bodonimhk

Bodonimhk

Answered question

2022-10-23

What will be the value of P ( 12 ) + P ( 8 ) if P ( x ) = x 4 + a x 3 + b x 2 + c x + d provided that P ( 1 ) = 10, P ( 2 ) = 20, P ( 3 ) = 30 ?

Answer & Explanation

Laci Conrad

Laci Conrad

Beginner2022-10-24Added 17 answers

Two remarks, to avoid almost every computation:
The polynomial P ( x ) 10 x has roots 1,2 and 3, hence there exists a polynomial Q such that P ( x ) 10 x = ( x 1 ) ( x 2 ) ( x 3 ) Q ( x )
The polynomial P ( x ) 10 x has degree 4 and leading coefficient 1, hence Q ( x ) = x + z for some unknown constant z whose value will be irrelevant.
Thus, P ( 12 ) + P ( 8 ) = 10 ( 12 8 ) + 11 10 9 ( 12 + z ) + 9 10 11 ( 8 z ), that is, P ( 12 ) + P ( 8 ) = 10 4 + 11 10 9 ( 12 + z + 8 z ) = 40 + 990 20 = 19840

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