# A Factor Theorem Question

#### 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
.....
.....
.....
.....

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

Staff Emeritus
Gold Member
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

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving