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I need a help for this question please.

$P$ is a polynomial defined by $$P(x)=ab\left(a-c\right)x^{3}+\left(a^{3}-a^{2}c+2ab^{2}-b^{2}c+abc\right)x^{2}+\left(2a^{2}b+b^{2}c+a^{2}c+b^{3}-abc\right)x+ab\left(b+c\right)$$ $$ a,b,c \in \mathbb{N}$$

  1. I already proved that $P(x)$ is divisible by $$abx^{2}+\left(a^{2}+b^{2}\right)x+ab$$ and I found that $$P(x) = (abx^{2}+\left(a^{2}+b^{2}\right)x+ab)\left(\left(a-c\right)x+\left(b+c\right)\right)$$

2. I need help for this question. I must prove then for $$x_{0}=\left(a+b+1\right)^{n}$$ $$ n \in \mathbb{N}$$ $$P(x_{0}) \text{ is divisible by } (a+b)^3$$

thanks

user376343
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cauchy
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  • have you tried mod arithmetic ? – mick Nov 12 '23 at 22:49
  • No, i Just try to factorise by (a+b)^3 – cauchy Nov 12 '23 at 23:10
  • Are you suppose to use certain ideas or are you free ? And/or do you want to use something specific ? I would personally use mod. Not sure if its the best though. But I think it will do fine. – mick Nov 12 '23 at 23:18

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