w^3 + 5w^2 - w = 5
w^3 + 5w^2 - w - 5 = 0
To solve this, we use the factor theorem
f(w) = w^3 + 5w^2 - w -5
If f(a) = 0, then (x-a) is a factor of f(x) --- Don't worry if you don't understand this, I didn't either when I did this for the first time a few years back.
Basically, we are substituting for values of x which will gives us 0 at the end.
f(0) = -5 NOT 0
f(1) = 1 + 5 -1 -5 = 0 --- This is what we want!
Hence, (w-1) is a factor of f(x), which is w^3 + 5w^2 - w -5 = 0
f(w) = w^3 + 5w^2 - w -5
Now, divide f(w) by (w-1). You can do this on paper but I've figured it out as:
f(w) = (w^2 + 6w + 5)(w-1)
Factorise the quadratic
(w+1)(w+5)(w-1) = 0
Hence, our solutions are 1,-1, -5
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