Factor completely.
$4 x^3-x^2-36 x+9$
We are given the cubic polynomial \(4x^3 - x^2 - 36x + 9\). We want to factor it completely. The polynomial has four terms, which suggests that factoring by grouping might be an effective strategy. We can group the terms as follows: \((4x^3 - x^2) + (-36x + 9)\).
Next, we factor out the greatest common factor from each group.
For the first group, \(4x^3 - x^2\), the greatest common factor is \(x^2\), so we can write this as \(x^2(4x - 1)\).
For the second group, \(-36x + 9\), the greatest common factor is \(-9\), so we can write this as \(-9(4x - 1)\).
Combining these, the expression becomes:
\(x^2(4x - 1) - 9(4x - 1)\).
Now that we have a common binomial factor of \((4x - 1)\), we can factor it out:
\((4x - 1)(x^2 - 9)\).
The expression \(x^2 - 9\) is a difference of squares, which follows the pattern \(a^2 - b^2 = (a - b)(a + b)\).
Thus, \(x^2 - 9 = (x - 3)(x + 3)\).
Combining all the factors, the original cubic polynomial can be written as \((4x - 1)(x - 3)(x + 3)\). This is the complete factorization of the expression. The final answer is \(\boxed{(4x - 1)(x - 3)(x + 3)}\).
Factoring polynomials like \(4x^3 - x^2 - 36x + 9\) is a fundamental skill in algebra with practical applications in fields such as engineering, physics, and computer science. For instance, when designing structural supports or analyzing the motion of an object, you often need to find the roots (or zeros) of a function to determine critical points or stability thresholds. By factoring the equation into \((4x - 1)(x - 3)(x + 3) = 0\), we can easily see that the zeros are \(x = 1/4\), \(x = 3\), and \(x = -3\), allowing engineers to quickly identify where the function crosses the axis or changes behavior.