Is (x + a + b)^7 - x^7 - a^7 - b^7 Divisible by x^2 + (a + b)x + ab?

[Hyperreality]

My maths teacher says this problem is not as impossible as it seems, but I just can't solve it.

Show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisble by
x^2 + (a + b)x +ab.

 

[drag]

Looking for the easy way out huh ?
You can always solve the entire excercise...

Live long and prosper.

 

[KLscilevothma]

Show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisble by x^2 + (a + b)x +ab.

Hint:
Notice that (x+a) and (x+b) are the 2 factors of x^2 + (a + b)x +ab.
So it is equivalent to show that (x + a + b)^7 - x^7 - a^7 - b^7 is divisible by both (x+a) and (x+b).

Let f(x) = (x + a + b)^7 - x^7 - a^7 - b^7
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Can you continue from here?

Hope this help.

 

[ottjes]

just write everything out
eg. (x+a)^2=x^2+2xa+a^2

maybe rewrite some terms then and you will see that it is divisible by x^2 + (a + b)x +ab

 

[Hurkyl]

KL has the easy way!

Writing it out however... *shudder* I wouldn't wish writing out a trinomial to the 7th power to anyone!

Hurkyl