To find the factored form of the expression [tex]\( x^3 - 6x^2 - 9x + 54 \)[/tex], let's break down the steps required to factor this polynomial:
1. Identify Possible Rational Roots:
First, we consider the possible rational roots of the polynomial using the Rational Root Theorem. However, let’s proceed directly to factoring, as it simplifies the process.
2. Group Terms:
Next, we group the terms to facilitate factoring by grouping:
[tex]\[
(x^3 - 6x^2) + (-9x + 54)
\][/tex]
3. Factor Common Terms:
Factor out the greatest common factor (GCF) from each group:
[tex]\[
x^2(x - 6) - 9(x - 6)
\][/tex]
4. Factor by Grouping:
Notice that [tex]\( (x - 6) \)[/tex] is a common factor:
[tex]\[
(x - 6)(x^2 - 9)
\][/tex]
5. Further Factorization:
Recognize that [tex]\( x^2 - 9 \)[/tex] is a difference of squares and can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
6. Combine All Factors:
Substituting these factors back, we get:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
Therefore, the factored form of the expression [tex]\( x^3 - 6x^2 - 9x + 54 \)[/tex] is:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
