Answer :
To factor the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], we begin by looking for any possible factors from the given options. The factored form derived is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex]. Let's go through the process step-by-step:
1. Look for Any Common Factors:
For the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], there are no common factors in all terms.
2. Use the Rational Root Theorem:
This allows us to test possible rational roots which are factors of the constant term, 54. The possible rational roots we could test are [tex]\(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Test Potential Roots:
Substituting these values into the polynomial helps to check if they make the polynomial equal zero. Testing [tex]\(x = 6\)[/tex], we find that it is a root as it makes the polynomial equal zero.
4. Factor Out the Root Found:
Since [tex]\(x = 6\)[/tex] is a root, we know [tex]\((x - 6)\)[/tex] is a factor. We can use synthetic division to divide the polynomial by [tex]\((x - 6)\)[/tex] to find the remaining factors.
5. Divide and Simplify:
After dividing by [tex]\((x - 6)\)[/tex], we get a quadratic polynomial: [tex]\(x^2 - 9\)[/tex].
6. Factor the Quadratic:
The quadratic [tex]\(x^2 - 9\)[/tex] is a difference of squares, which factors into [tex]\((x - 3)(x + 3)\)[/tex].
7. Combine All Factors:
Combining all the factors found, the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] can be factored as [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
Thus, the factored form of the expression is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
1. Look for Any Common Factors:
For the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], there are no common factors in all terms.
2. Use the Rational Root Theorem:
This allows us to test possible rational roots which are factors of the constant term, 54. The possible rational roots we could test are [tex]\(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Test Potential Roots:
Substituting these values into the polynomial helps to check if they make the polynomial equal zero. Testing [tex]\(x = 6\)[/tex], we find that it is a root as it makes the polynomial equal zero.
4. Factor Out the Root Found:
Since [tex]\(x = 6\)[/tex] is a root, we know [tex]\((x - 6)\)[/tex] is a factor. We can use synthetic division to divide the polynomial by [tex]\((x - 6)\)[/tex] to find the remaining factors.
5. Divide and Simplify:
After dividing by [tex]\((x - 6)\)[/tex], we get a quadratic polynomial: [tex]\(x^2 - 9\)[/tex].
6. Factor the Quadratic:
The quadratic [tex]\(x^2 - 9\)[/tex] is a difference of squares, which factors into [tex]\((x - 3)(x + 3)\)[/tex].
7. Combine All Factors:
Combining all the factors found, the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] can be factored as [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
Thus, the factored form of the expression is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
