150 Followers Question

Algebra Level 3

Let P ( x ) = x 5 + a x 4 + b x 3 + c x 2 + d x + e P(x) = x^{5}+ax^{4}+bx^{3}+cx^{2}+dx+e be a polynomial with real coefficients and satisfying the property P ( n ) = 10 n P(n) = 10n for n = 1 , 2 , 3 , 4 , n = 1, 2, 3, 4, and 5 5 . Find the value of a + b + c + d + e . a + b + c + d + e.


The answer is 9.

Benedict Dimacutac by Benedict Dimacutac , 21, Philippines

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2 solutions

Mohtasim Nakib
Jan 26, 2016

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Let g ( x ) = P ( x ) − 10 x g(x)=P(x)-10x .Then g ( x ) = 0 g(x)=0 for x = 1 , 2 , 3 , 4 , 5 x=1,2,3,4,5 .As P ( x ) P(x) is monic,therefore g ( x ) g(x) is monic as well.

Therefore, g ( x ) g(x) can be represented as: ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) ( x − 5 ) = P ( x ) − 10 x ⟹ P ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) ( x − 5 ) + 10 x ⟹ P ( x ) = x 5 − 15 x 4 + 85 x 3 − 225 x 2 + 284 x − 120 ⟹ a = − 15 , b = 85 , c = − 225 , d = 284 , e = − 120 ⟹ a + b + c + d + e = 9 (x-1)(x-2)(x-3)(x-4)(x-5)=P(x)-10x\\ \implies P(x)=(x-1)(x-2)(x-3)(x-4)(x-5)+10x\\ \implies P(x)=x^5-15x^4+85x^3-225x^2+284x-120\\ \implies a=-15,b=85,c=-225,d=284,e=-120\\ \implies \boxed{a+b+c+d+e=9}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Instead of expanding the polynomial, we can substitute in x = − 1 x = - 1 .

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