Answer :
To factor the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], we need to find the factors that multiply together to give the original polynomial.
1. Grouping the Terms: We start by considering grouping if it might be helpful in finding common factors.
Group the expression as:
[tex]\[
(x^3 - 6x^2) + (-9x + 54)
\][/tex]
2. Factoring by Grouping: In the first group, [tex]\(x^3 - 6x^2\)[/tex], factor out the greatest common factor, which is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x - 6)
\][/tex]
In the second group, [tex]\(-9x + 54\)[/tex], factor out [tex]\(-9\)[/tex]:
[tex]\[
-9(x - 6)
\][/tex]
3. Combine the Groups: Now, notice that both groups contain [tex]\((x - 6)\)[/tex] as a common factor:
[tex]\[
x^2(x - 6) - 9(x - 6) = (x^2 - 9)(x - 6)
\][/tex]
4. Factor Further if Possible: Now, look at the factor [tex]\(x^2 - 9\)[/tex]. This is a difference of squares, which can be factored as:
[tex]\[
(x + 3)(x - 3)
\][/tex]
5. Final Factored Form: Putting it all together, the expression becomes:
[tex]\[
(x - 6)(x + 3)(x - 3)
\][/tex]
So, the completely factored form of the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
1. Grouping the Terms: We start by considering grouping if it might be helpful in finding common factors.
Group the expression as:
[tex]\[
(x^3 - 6x^2) + (-9x + 54)
\][/tex]
2. Factoring by Grouping: In the first group, [tex]\(x^3 - 6x^2\)[/tex], factor out the greatest common factor, which is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x - 6)
\][/tex]
In the second group, [tex]\(-9x + 54\)[/tex], factor out [tex]\(-9\)[/tex]:
[tex]\[
-9(x - 6)
\][/tex]
3. Combine the Groups: Now, notice that both groups contain [tex]\((x - 6)\)[/tex] as a common factor:
[tex]\[
x^2(x - 6) - 9(x - 6) = (x^2 - 9)(x - 6)
\][/tex]
4. Factor Further if Possible: Now, look at the factor [tex]\(x^2 - 9\)[/tex]. This is a difference of squares, which can be factored as:
[tex]\[
(x + 3)(x - 3)
\][/tex]
5. Final Factored Form: Putting it all together, the expression becomes:
[tex]\[
(x - 6)(x + 3)(x - 3)
\][/tex]
So, the completely factored form of the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
