To find the complete factorization of the polynomial ( x^3 + 4x^2 - x - 4 ), we look for factors that will multiply together to give us the original polynomial. We can sometimes guess at one of the factors by using the Rational Root Theorem or synthetic division, or by simply observing patterns.
Let's try to factor by grouping, where we group terms together that have common factors:
( x^3 + 4x^2 - x - 4 )
This can be regrouped as:
( (x^3 - x) + (4x^2 - 4) )
Now, we can factor out the greatest common factor from each group:
( x(x^2 - 1) + 4(x^2 - 1) )
Notice that ( x^2 - 1 ) is common to both terms, we can factor that out:
(x + 4)(x^2 - 1)
The term ( x^2 - 1 ) is a difference of squares, which can be further factored:
( x^2 - 1 = (x - 1)(x + 1) )
So by substituting back, the complete factorization is:
( (x + 4)(x - 1)(x + 1) )
These are the factors of the given polynomial.