To factor the polynomial [tex]\(x^2 - 3x - 10\)[/tex], we look for two numbers that multiply to the constant term, [tex]\(-10\)[/tex], and add up to the coefficient of the linear term, [tex]\(-3\)[/tex].
1. We need integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
[tex]\[
a \cdot b = -10 \quad \text{and} \quad a + b = -3.
\][/tex]
2. Listing the factors of [tex]\(-10\)[/tex], we have:
[tex]\[
(-1, 10), (1, -10), (-2, 5), (2, -5).
\][/tex]
3. We check the sum of each pair to see which equals [tex]\(-3\)[/tex]:
- [tex]\((-1) + 10 = 9\)[/tex]
- [tex]\(1 + (-10) = -9\)[/tex]
- [tex]\((-2) + 5 = 3\)[/tex]
- [tex]\(2 + (-5) = -3\)[/tex]
4. The pair that sums to [tex]\(-3\)[/tex] is [tex]\(2\)[/tex] and [tex]\(-5\)[/tex].
5. We can now write the polynomial as:
[tex]\[
x^2 - 3x - 10 = (x + 2)(x - 5)
\][/tex]
Therefore, the polynomial [tex]\(x^2 - 3x - 10\)[/tex] factors completely as [tex]\((x + 2)(x - 5)\)[/tex].
The correct choice is:
A. [tex]\[
x^2 - 3x - 10 = (x + 2)(x - 5)
\][/tex]
