Factor completely

x^3 - 5x^2 - x + 5

looks like we can use grouping

x^3 - 5x^2 - x + 5
= x^2(x-5) - 1(x-5)
= (x-5)(x^2-1)
= (x-5)(x+1)(x-1)

Thank you!

To factor the polynomial completely, we should try to find any common factors among the terms. Then, we can apply factoring techniques such as grouping or factoring by grouping, if applicable.

In this case, let's see if there are any common factors among the terms. The greatest common factor (GCF) of the given terms, x^3 - 5x^2 - x + 5, is 1, as we cannot factor out any common variable or numeric factor.

Now, let's try factoring by grouping. We can group the terms in pairs so that we can factor out a common factor from each pair.

x^3 - 5x^2 - x + 5

= (x^3 - 5x^2) + (-x + 5)

= x^2(x - 5) - 1(x - 5)

Notice that we now have a common factor of (x - 5) in both terms after grouping. We can factor it out:

= (x - 5)(x^2 - 1)

Using the difference of squares identity, x^2 - 1 = (x + 1)(x - 1), we can further simplify the expression:

= (x - 5)(x + 1)(x - 1)

Therefore, the completely factored form of the given polynomial is (x - 5)(x + 1)(x - 1).