To factor the polynomial [tex]\(x^3 + 6x^2 - 9x - 54\)[/tex] completely, let's go through the process step-by-step.
Step 1: Recognize a Potential Rational Root
First, we can use the Rational Root Theorem to list possible rational roots of the polynomial. For [tex]\(ax^n + bx^{n-1} + ... + k\)[/tex], rational roots are factors of the constant term ([tex]\(k = -54\)[/tex]) divided by factors of the leading coefficient ([tex]\(a = 1\)[/tex]).
The possible rational roots are:
[tex]\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54
\][/tex]
Step 2: Test the Potential Roots
We can substitute these values into the polynomial to check if they are zero. Let's start by testing [tex]\(x = 3\)[/tex]:
[tex]\[
3^3 + 6(3^2) - 9(3) - 54 = 27 + 54 - 27 - 54 = 0
\][/tex]
[tex]\(x = 3\)[/tex] is a root.
Step 3: Use Synthetic Division
Knowing that [tex]\(x = 3\)[/tex] is a root, we can use synthetic division to factor out [tex]\((x - 3)\)[/tex]:
[tex]\[
\begin{array}{r|rrrr}
3 & 1 & 6 & -9 & -54 \\
& & 3 & 27 & 54 \\
\hline
& 1 & 9 & 18 & 0 \\
\end{array}
\][/tex]
The quotient from synthetic division is [tex]\(x^2 + 9x + 18\)[/tex]. Thus, we have:
[tex]\[
x^3 + 6x^2 - 9x - 54 = (x - 3)(x^2 + 9x + 18)
\][/tex]
Step 4: Factor the Quadratic
Next, we factor [tex]\(x^2 + 9x + 18\)[/tex]. We can use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] to find the roots:
[tex]\[
b = 9, \quad a = 1, \quad c = 18
\][/tex]
[tex]\[
x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-9 \pm \sqrt{81 - 72}}{2}
\][/tex]
[tex]\[
x = \frac{-9 \pm 3}{2}
\][/tex]
[tex]\[
x = \frac{-6}{2} = -3 \quad \text{and} \quad x = \frac{-12}{2} = -6
\][/tex]
Thus, the factors of the quadratic are:
[tex]\[
x^2 + 9x + 18 = (x + 3)(x + 6)
\][/tex]
Step 5: Write the Complete Factorization
Combining these results, the complete factorization of the polynomial [tex]\(x^3 + 6x^2 - 9x - 54\)[/tex] is:
[tex]\[
x^3 + 6x^2 - 9x - 54 = (x - 3)(x + 3)(x + 6)
\][/tex]
This matches one of the given options:
[tex]\[
(x + 3)(x - 3)(x + 6)
\][/tex]
Hence, the correct factorization of the polynomial is:
[tex]\[
(x + 3)(x - 3)(x + 6)
\][/tex]
