Polynomial

[MathBuff]

I need a help with this problem

 

[MarkFL]

I would begin by observing that:

[MATH]x^3-1=(x-1)(x^2+x+1)[/MATH]
And so we know the roots of \(x^2+x+1\) are complex cube roots of unity.

Can you proceed?

 

[MathBuff]

Can i get more help? Cause i still don't know, sorry

 

[firemath]

@MathBuff, you need to at least try what MarkFL has given you. Don't give up yet!

 

[MarkFL]

Can i get more help? Cause i still don't know, sorry

Suppose \(\beta\) is a complex cube root of unity. Then we must have:

[MATH]\beta^{3k}=(\beta^3)^k=1^k=1[/MATH] where \(k\in\mathbb{N}\).

And since \(f(x)\) is divisible by \(x^2+x+1\) what must \(f(\beta)\) be?

Hint: [MATH]f(x)=(x^2+x+1)P(x)[/MATH]