# factoring question

• January 30th 2013, 05:33 AM
Lotrnerd
factoring question
Could someone help me with the following question?

a^3 + a^2 - a - 1

I can't seem so get to the answer. Id love to know how you arrive to the answer too.

Thanks
• January 30th 2013, 06:39 AM
Shakarri
Re: factoring question
Using the factor theorem the roots multiplied together have to be -1 or 1 (the constant term in the equation)

This is a small proof od the factor theorem.
Quote:

The cubic equation should have three roots, call these roots p, q and r
When these are the three roots
(a-p)(a-q)(a-r)= a^3 + a^2 - a - 1
a^3 - (p+q+r)*a^2 + (pq+pr+rq)*a - pqr= a^3 + a^2 - a - 1
The constant on the left hand side -pqr is equal to the constant on the right hand side -1
So pqr= 1
Whatever the values of p, q and r are they must multiply to give 1.
It is possible that they are not integers (whole numbers) but to begin with assume that they are.
The only factors of 1 are 1 and -1
Test if a=1 is a root
(1)^3 + (1)^2 - (1) -1= 0

So 1 is a root.
a=1 is a root implies (a-1) is a factor.

use long division to divide a^3 + a^2 - a - 1 by (a-1) and you will get a quadratic equation. Solve the quadratic equation to find the other two roots.
• January 30th 2013, 06:57 AM
Lotrnerd
Re: factoring question
Thank you for the help. I undertand the first part, I just don't know how to do long division in this situation to get the last two roots.
• January 30th 2013, 07:00 AM
Plato
Re: factoring question
Quote:

Originally Posted by Lotrnerd
Could someone help me with the following question?
a^3 + a^2 - a - 1
I can't seem so get to the answer. Id love to know how you arrive to the answer too.

$a^3+a^2-a-1\\a^2(a+1)-(a+1)\\(a+1)(a^2-1)\\(a+1)^2(a-1)$
• January 31st 2013, 04:38 AM
Shakarri
Re: factoring question
Plato's method of factorising directly works but it wont always be that straight forward. Long dividing will get the same answer but will always work.
Quote:

Originally Posted by Lotrnerd
Thank you for the help. I undertand the first part, I just don't know how to do long division in this situation to get the last two roots.

Its hard to write long division on these forums, take a look at this example Algebraic Long Division