To factorize the trinomial [tex]\(x^2 - 3x - 18\)[/tex], we need to find two binomials that multiply together to give the original quadratic expression.
The general approach for factorizing a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex] involves finding two numbers that:
1. Multiply to give the constant term ([tex]\(c\)[/tex]).
2. Add to give the coefficient of the middle term ([tex]\(b\)[/tex]).
For the trinomial [tex]\(x^2 - 3x - 18\)[/tex], the constant term [tex]\(c\)[/tex] is [tex]\(-18\)[/tex] and the coefficient of the middle term [tex]\(b\)[/tex] is [tex]\(-3\)[/tex].
We need to find two numbers that multiply to [tex]\(-18\)[/tex] and add up to [tex]\(-3\)[/tex]. After checking various pairs of factors of [tex]\(-18\)[/tex]:
- [tex]\(9\)[/tex] and [tex]\(-2\)[/tex] do not work because [tex]\(9 + (-2) = 7\)[/tex].
- [tex]\(-9\)[/tex] and [tex]\(2\)[/tex] do not work because [tex]\(-9 + 2 = -7\)[/tex].
- [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] work because [tex]\(-6 + 3 = -3\)[/tex].
Thus, the numbers we are looking for are [tex]\(-6\)[/tex] and [tex]\(3\)[/tex].
So, we can factorize our trinomial as:
[tex]\[
x^2 - 3x - 18 = (x - 6)(x + 3)
\][/tex]
This matches option D. Therefore, the correct factorization of [tex]\(x^2 - 3x - 18\)[/tex] is:
[tex]\[
(x - 6)(x + 3)
\][/tex]
