Final answer:
To factor the expression 9x^3 - 36x^2 - x - 4 completely, you can use various methods such as grouping, difference of squares, and trinomial factoring.
Step-by-step explanation:
To factor the expression 9x^3 - 36x^2 - x - 4 completely, you can use various methods such as grouping, difference of squares, and trinomial factoring. In this case, a possible method is trinomial factoring.
First, identify if the expression is a perfect square trinomial. Checking the first and last terms, 9x^3 and 4, we can see that they are not perfect squares. Therefore, we need to look for factors that add up to the middle term, -x. The factors are -3x and 3x. Now we can rewrite the expression as (9x^3 - 3x) + (-36x^2 + 12x) - (x - 4).
Next, factor out the greatest common factor (GCF) from each binomial. The GCF of the first binomial is 3x, and the GCF of the second binomial is 12. Factoring them out, we get 3x(3x^2 - 1) + 12(-3x^2 + 4x) - (x - 4).
Simplifying further, we have 3x(3x^2 - 1) - 12(3x^2 - 4x) - (x - 4).
Now, we can factor out the common binomial, 3x - 4, from the remaining terms. This gives us (3x - 4)(3x^2 - 1) - (x - 4).
Finally, we can factor out the common binomial, x - 4, from the expression. This results in (3x - 4)(3x^2 - 1)(x - 4).