Answer :
Sure, let's break down how to factor the given expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] step-by-step.
1. Identify Possible Rational Roots: First, we should look for possible values of [tex]\(x\)[/tex] that make the polynomial equal to zero. These values are known as the roots of the polynomial.
2. Synthetic Division: Use synthetic division to test possible roots derived from the Rational Root Theorem. The Rational Root Theorem suggests that the possible rational roots of the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] are factors of the constant term (54) divided by factors of the leading coefficient (1). These possible roots include [tex]\(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Finding a Root: We test these roots one-by-one. Let's consider [tex]\(x = 6\)[/tex].
- Substitute [tex]\(x = 6\)[/tex] into the polynomial:
[tex]\( (6)^3 - 6(6)^2 - 9(6) + 54 \)[/tex]
- This simplifies to:
[tex]\( 216 - 216 - 54 + 54 = 0 \)[/tex]
- Since [tex]\(x = 6\)[/tex] is a root, [tex]\(x - 6\)[/tex] is a factor of the polynomial.
4. Factor out [tex]\(x - 6\)[/tex]: Next, we divide [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] by [tex]\(x - 6\)[/tex] to find the quotient polynomial.
- Perform the division:
[tex]\[
\begin{array}{r|rrrr}
& x^3 & -6x^2 & -9x & +54 \\
6 & 1 & -6 & -9 & +54 \\
\hline
& 1 & 0 & -9 & +0 \\
\end{array}
\][/tex]
- This division yields [tex]\(x^2 - 3x - 9\)[/tex].
5. Factor the quadratic polynomial: The quotient polynomial we obtained is [tex]\(x^2 - 3x - 9\)[/tex].
- We can factor this quadratic expression:
[tex]\((x - 3)(x + 3)\)[/tex]
6. Combine all factors: Now, combining with the previously found factor, we get:
[tex]\[ x^3 - 6x^2 - 9x + 54 = (x - 6)(x - 3)(x + 3) \][/tex]
Therefore, the factored form of [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
So, the factors [tex]\(x - 6\)[/tex], [tex]\(x - 3\)[/tex], and [tex]\(x + 3\)[/tex] match the given options.
1. Identify Possible Rational Roots: First, we should look for possible values of [tex]\(x\)[/tex] that make the polynomial equal to zero. These values are known as the roots of the polynomial.
2. Synthetic Division: Use synthetic division to test possible roots derived from the Rational Root Theorem. The Rational Root Theorem suggests that the possible rational roots of the polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] are factors of the constant term (54) divided by factors of the leading coefficient (1). These possible roots include [tex]\(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Finding a Root: We test these roots one-by-one. Let's consider [tex]\(x = 6\)[/tex].
- Substitute [tex]\(x = 6\)[/tex] into the polynomial:
[tex]\( (6)^3 - 6(6)^2 - 9(6) + 54 \)[/tex]
- This simplifies to:
[tex]\( 216 - 216 - 54 + 54 = 0 \)[/tex]
- Since [tex]\(x = 6\)[/tex] is a root, [tex]\(x - 6\)[/tex] is a factor of the polynomial.
4. Factor out [tex]\(x - 6\)[/tex]: Next, we divide [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] by [tex]\(x - 6\)[/tex] to find the quotient polynomial.
- Perform the division:
[tex]\[
\begin{array}{r|rrrr}
& x^3 & -6x^2 & -9x & +54 \\
6 & 1 & -6 & -9 & +54 \\
\hline
& 1 & 0 & -9 & +0 \\
\end{array}
\][/tex]
- This division yields [tex]\(x^2 - 3x - 9\)[/tex].
5. Factor the quadratic polynomial: The quotient polynomial we obtained is [tex]\(x^2 - 3x - 9\)[/tex].
- We can factor this quadratic expression:
[tex]\((x - 3)(x + 3)\)[/tex]
6. Combine all factors: Now, combining with the previously found factor, we get:
[tex]\[ x^3 - 6x^2 - 9x + 54 = (x - 6)(x - 3)(x + 3) \][/tex]
Therefore, the factored form of [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is:
[tex]\[
(x - 6)(x - 3)(x + 3)
\][/tex]
So, the factors [tex]\(x - 6\)[/tex], [tex]\(x - 3\)[/tex], and [tex]\(x + 3\)[/tex] match the given options.
