Answer :
To factor the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], let's break it down step-by-step.
1. Identify potential rational roots: According to the Rational Root Theorem, the possible rational roots are factors of the constant term (54) divided by factors of the leading coefficient (1). These factors are [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54 \)[/tex].
2. Test the roots: Begin checking these potential roots using synthetic division or direct substitution into the polynomial to find if any are actual roots.
3. Find one real root: By testing, we'll discover that [tex]\( x = 6 \)[/tex] is a root because substituting [tex]\( x = 6 \)[/tex] into the polynomial yields zero:
[tex]\[
6^3 - 6 \times 6^2 - 9 \times 6 + 54 = 0
\][/tex]
This means [tex]\( x - 6 \)[/tex] is a factor.
4. Divide the polynomial: Use synthetic division to divide the original polynomial by [tex]\( x - 6 \)[/tex]. You should get:
[tex]\[
x^3 - 6x^2 - 9x + 54 = (x - 6)(x^2 - 9)
\][/tex]
5. Factor the quadratic: The quadratic [tex]\( x^2 - 9 \)[/tex] is a difference of squares and can be factored further into:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
6. Combine all the factors: Now combine all these factors together to get the complete factored form:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
So, the factored form of the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
1. Identify potential rational roots: According to the Rational Root Theorem, the possible rational roots are factors of the constant term (54) divided by factors of the leading coefficient (1). These factors are [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54 \)[/tex].
2. Test the roots: Begin checking these potential roots using synthetic division or direct substitution into the polynomial to find if any are actual roots.
3. Find one real root: By testing, we'll discover that [tex]\( x = 6 \)[/tex] is a root because substituting [tex]\( x = 6 \)[/tex] into the polynomial yields zero:
[tex]\[
6^3 - 6 \times 6^2 - 9 \times 6 + 54 = 0
\][/tex]
This means [tex]\( x - 6 \)[/tex] is a factor.
4. Divide the polynomial: Use synthetic division to divide the original polynomial by [tex]\( x - 6 \)[/tex]. You should get:
[tex]\[
x^3 - 6x^2 - 9x + 54 = (x - 6)(x^2 - 9)
\][/tex]
5. Factor the quadratic: The quadratic [tex]\( x^2 - 9 \)[/tex] is a difference of squares and can be factored further into:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
6. Combine all the factors: Now combine all these factors together to get the complete factored form:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
So, the factored form of the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
