To factor the polynomial function \( b(x) = x^3 + 2x^2 - 9x - 18 \), we follow these steps:
1. Identify the Polynomial:
We start with the polynomial \( b(x) = x^3 + 2x^2 - 9x - 18 \).
2. Find Rational Roots:
Using the Rational Root Theorem, we consider all possible rational roots of the polynomial, which are the factors of the constant term (-18) divided by the factors of the leading coefficient (1). These factors are:
[tex]\[
\text{Potential Roots:} \quad \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18
\][/tex]
Let's test these roots by substituting them into the polynomial until we find a root that results in the polynomial equating to zero.
3. Test Roots (picked conveniently for the solution):
- Testing \( x = 3 \):
[tex]\[
b(3) = 3^3 + 2(3^2) - 9(3) - 18 = 27 + 18 - 27 - 18 = 0
\][/tex]
Since \( b(3) = 0 \), \( x = 3 \) is a root.
4. Factor Out the Root:
Since \( x = 3 \) is a root, we can factor out \( (x - 3) \). We then perform polynomial division to divide \( b(x) \) by \( (x - 3) \):
[tex]\[
\frac{x^3 + 2x^2 - 9x - 18}{x - 3}
\][/tex]
Performing the division, we get:
[tex]\[
x^2 + 5x + 6
\][/tex]
5. Factor the Quotient \( x^2 + 5x + 6 \):
We now factor \( x^2 + 5x + 6 \) which can be factored into:
[tex]\[
(x + 2)(x + 3)
\][/tex]
6. Write the Fully Factored Form:
Combining all the factors, the fully factored form of the polynomial is:
[tex]\[
b(x) = (x - 3)(x + 2)(x + 3)
\][/tex]
Thus, the factored form of the polynomial [tex]\( b(x) = x^3 + 2x^2 - 9x - 18 \)[/tex] is [tex]\((x - 3)(x + 2)(x + 3)\)[/tex].
