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Answer :
To factor the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], we can follow these steps:
1. Look for Common Factors:
Check if there is a common factor in all the terms. In this case, there isn't any common factor for all terms other than 1. So, we will proceed with other methods.
2. Checking for Possible Rational Roots:
Use the Rational Root Theorem to find possible rational roots. For [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], the possible rational roots could be the factors of the constant term (54) divided by the factors of the leading coefficient (1). These are [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Test Possible Roots:
Substitute these values into the polynomial to find a root.
- [tex]\(x = 3\)[/tex]:
[tex]\((3)^3 - 6(3)^2 - 9(3) + 54 = 27 - 54 - 27 + 54 = 0\)[/tex].
Since [tex]\(x = 3\)[/tex] is a root, we know that [tex]\(x - 3\)[/tex] is a factor of the polynomial.
4. Use Polynomial Division:
Divide the original polynomial by [tex]\(x - 3\)[/tex] to find the other factors.
Dividing [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] by [tex]\(x - 3\)[/tex] gives us a quotient of [tex]\(x^2 - 3x - 18\)[/tex].
5. Factor the Quadratic:
Now, we need to factor the quadratic expression [tex]\(x^2 - 3x - 18\)[/tex].
Look for two numbers that multiply to [tex]\(-18\)[/tex] and add to [tex]\(-3\)[/tex]. The numbers [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] fit.
So, [tex]\(x^2 - 3x - 18 = (x - 6)(x + 3)\)[/tex].
6. Combine the Factors:
Therefore, the factored form of the original polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 3)(x - 6)(x + 3)\)[/tex].
So, the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] factors as [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
1. Look for Common Factors:
Check if there is a common factor in all the terms. In this case, there isn't any common factor for all terms other than 1. So, we will proceed with other methods.
2. Checking for Possible Rational Roots:
Use the Rational Root Theorem to find possible rational roots. For [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex], the possible rational roots could be the factors of the constant term (54) divided by the factors of the leading coefficient (1). These are [tex]\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm 27, \pm 54\)[/tex].
3. Test Possible Roots:
Substitute these values into the polynomial to find a root.
- [tex]\(x = 3\)[/tex]:
[tex]\((3)^3 - 6(3)^2 - 9(3) + 54 = 27 - 54 - 27 + 54 = 0\)[/tex].
Since [tex]\(x = 3\)[/tex] is a root, we know that [tex]\(x - 3\)[/tex] is a factor of the polynomial.
4. Use Polynomial Division:
Divide the original polynomial by [tex]\(x - 3\)[/tex] to find the other factors.
Dividing [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] by [tex]\(x - 3\)[/tex] gives us a quotient of [tex]\(x^2 - 3x - 18\)[/tex].
5. Factor the Quadratic:
Now, we need to factor the quadratic expression [tex]\(x^2 - 3x - 18\)[/tex].
Look for two numbers that multiply to [tex]\(-18\)[/tex] and add to [tex]\(-3\)[/tex]. The numbers [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] fit.
So, [tex]\(x^2 - 3x - 18 = (x - 6)(x + 3)\)[/tex].
6. Combine the Factors:
Therefore, the factored form of the original polynomial [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] is [tex]\((x - 3)(x - 6)(x + 3)\)[/tex].
So, the expression [tex]\(x^3 - 6x^2 - 9x + 54\)[/tex] factors as [tex]\((x - 6)(x - 3)(x + 3)\)[/tex].
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