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.
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.
The constant on the left hand side -pqr is equal to the constant on the right hand side -1The 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
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.
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.
Algebraic Long Division