What is the completely factored form of $x^3+4 x^2-9 x-36$ ?
We are tasked with finding the completely factored form of the cubic polynomial \(p(x) = x^3 + 4x^2 - 9x - 36\). A polynomial is considered "completely factored" when it is represented as the product of its irreducible factors. In this case, we have four terms, which suggests that factoring by grouping is an appropriate technique. Our goal is to manipulate the expression to reveal common factors among the terms.
To factor the polynomial by grouping, we divide the four terms into two pairs: the first two terms and the last two terms.
We group the expression as follows:
\[p(x) = (x^3 + 4x^2) + (-9x - 36)\]
Next, we factor out the greatest common factor (GCF) from each group.
From the first group \((x^3 + 4x^2)\), we can factor out \(x^2\):
\[x^2(x + 4)\]
From the second group \((-9x - 36)\), we can factor out \(-9\):
\[-9(x + 4)\]
Combining these, the expression becomes:
\[p(x) = x^2(x + 4) - 9(x + 4)\]
Now that we have rewritten the expression as \(x^2(x + 4) - 9(x + 4)\), we can clearly see the common binomial factor of \((x + 4)\).
Factoring out this common binomial:
\[p(x) = (x^2 - 9)(x + 4)\]
To ensure the polynomial is completely factored, we must check if the remaining quadratic factor \((x^2 - 9)\) can be factored further. We recognize \(x^2 - 9\) as a difference of squares, which follows the identity \(a^2 - b^2 = (a - b)(a + b)\).
With \(a = x\) and \(b = 3\), we have:
\[x^2 - 3^2 = (x - 3)(x + 3)\]
Substituting this back into our expression, we obtain:
\[p(x) = (x - 3)(x + 3)(x + 4)\]
Comparing our result \((x - 3)(x + 3)(x + 4)\) with the provided options:
We see that option (a) is equivalent to our derived form. Option (d) contains a factor that can be reduced further, so it is not completely factored. Thus, the correct choice is (a), and the completely factored form is \(\boxed{(x+3)(x-3)(x+4)}\).
Factoring polynomials is a fundamental tool in algebra used for simplifying expressions, solving equations, and graphing functions. For example, in engineering, when analyzing the stability of a system, one often needs to find the "roots" of a characteristic polynomial. By completely factoring the polynomial, one can easily identify these roots, which helps in understanding the system's behavior and performance. This process is essential for designing robust systems in fields ranging from structural engineering to control systems design.