A glance at a problem

Algebra Level 3

Find a a so that x + 2 ∣ 2 x 3 − 3 x 2 + x + a x + 2 \ | \ 2x^3 - 3x^2 + x + a .

This is part of the series: It's easy, believe me!


The answer is 30.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

Marco Brezzi
Aug 13, 2017

If x + 2 ∣ f ( x ) x+2|f(x) where f ( x ) = 2 x 3 − 3 x 2 + x + a f(x)=2x^3-3x^2+x+a , x + 2 x+2 must be a factor of f ( x ) f(x) or equivalently

f ( − 2 ) = 0 ⟺ 2 ( − 2 ) 3 − 3 ( − 2 ) 2 − 2 + a = 0 ⟺ a = 30 \begin{aligned} f(-2)=0&\iff 2(-2)^3-3(-2)^2-2+a=0\\ &\iff a=\boxed{30} \end{aligned}

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