Christmas Streak 34/88: Seems Simple Enough!

Algebra Level 4

A cubic polynomial P ( x ) P(x) satisfies the below conditions:

  • ( x − 1 ) P ( x − 2 ) = ( x − 7 ) P ( x ) . (x-1)P(x-2)=(x-7)P(x).

  • The remainder when P ( x ) P(x) is divided by ( x 2 − 4 x + 2 ) (x^2-4x+2) is 2 x − 10. 2x-10.

Find the value of P ( 4 ) . P(4).

This problem is a part of <Christmas Streak 2017> series .

-6 0 6 3 -3

Boi (보이) by Boi (보이) , 19, South Korea

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2 solutions

Boi (보이)
Nov 1, 2017

Relevant wiki: Polynomial Division

Answer: -6

Substitute x = 1 , 7 x=1,~7 to the first condition, and we see that P ( 1 ) = P ( 5 ) = 0. P(1)=P(5)=0.

Then since P ( x ) = ( x 2 − 4 x + 2 ) ( a x + b ) + 2 x − 10 , P(x)=(x^2-4x+2)(ax+b)+2x-10, we substitute x = 1 , 5. x=1,~5.

{ a + b = − 8 5 a + b = 0 ⇒ a = 2 , b = − 10. \cases{a+b=-8 \\\\ 5a+b=0}~\Rightarrow~a=2,~b=-10.

∴ P ( x ) = ( x 2 − 4 x + 2 ) ( 2 x − 10 ) + 2 x − 10 = ( x 2 − 4 x + 3 ) ( 2 x − 10 ) = 2 ( x − 1 ) ( x − 3 ) ( x − 5 ) . \therefore~P(x)=(x^2-4x+2)(2x-10)+2x-10=(x^2-4x+3)(2x-10)=2(x-1)(x-3)(x-5).

P ( 4 ) = 2 ⋅ 3 ⋅ 1 ⋅ ( − 1 ) = − 6 . P(4)=2\cdot 3\cdot 1\cdot (-1)=\boxed{-6}.

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Atvthe King
Jul 21, 2020

Since P is a cubic, and that 3x2 = 6, we realize that we can write P ( x ) = a ( x − 1 ) ( x − 3 ) ( x − 5 ) P(x) = a(x-1)(x-3)(x-5) . The remainder upon division by the given quadratic is x − 5 x-5 , so a = 2 a = 2 . The answer is then 6.

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