To determine the factors of the quadratic polynomial [tex]\( x^2 - 3x - 18 \)[/tex], we follow these steps:
1. Recognize the quadratic polynomial in standard form:
[tex]\[
ax^2 + bx + c
\][/tex]
For our polynomial, [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -18 \)[/tex].
2. We seek two numbers that multiply to [tex]\( ac \)[/tex] (which is [tex]\( 1 \cdot -18 = -18 \)[/tex]) and add up to [tex]\( b \)[/tex] (which is [tex]\(-3\)[/tex]).
3. These two numbers are [tex]\( -6 \)[/tex] and [tex]\( 3 \)[/tex] because:
- [tex]\(-6 \times 3 = -18\)[/tex]
- [tex]\(-6 + 3 = -3\)[/tex]
4. Rewrite the middle term ([tex]\(-3x\)[/tex]) of the polynomial using the numbers [tex]\(-6\)[/tex] and [tex]\( 3 \)[/tex]:
[tex]\[
x^2 - 6x + 3x - 18
\][/tex]
5. Group the terms in pairs:
[tex]\[
(x^2 - 6x) + (3x - 18)
\][/tex]
6. Factor out the greatest common factor from each pair:
[tex]\[
x(x - 6) + 3(x - 6)
\][/tex]
7. Notice that [tex]\((x - 6)\)[/tex] is a common factor:
[tex]\[
(x - 6)(x + 3)
\][/tex]
Therefore, the factors of the polynomial [tex]\( x^2 - 3x - 18 \)[/tex] are [tex]\((x - 6)\)[/tex] and [tex]\((x + 3)\)[/tex].
Thus, the correct answer is:
A [tex]\((x - 6), (x + 3)\)[/tex]
