To factor the given trinomial [tex]\(x^2 - 3x - 10\)[/tex], follow these detailed steps:
1. Identify the coefficients: The given trinomial is [tex]\(x^2 - 3x - 10\)[/tex]. Here, the coefficient of [tex]\(x^2\)[/tex] (the leading coefficient) is 1, the coefficient of [tex]\(x\)[/tex] (the linear term) is -3, and the constant term is -10.
2. Find two numbers that multiply to the constant term (-10) and add to the linear coefficient (-3). These two numbers will be used to split the middle term:
[tex]\[
\text{The pair of numbers is } (-5) \text{ and } (+2)
\][/tex]
because:
[tex]\[
(-5) \times (+2) = -10 \quad \text{and} \quad (-5) + (+2) = -3
\][/tex]
3. Rewrite the middle term (-3x) using these numbers:
[tex]\[
x^2 - 5x + 2x - 10
\][/tex]
4. Factor by grouping: Group the terms in pairs and factor out the common factors from each pair:
[tex]\[
x^2 - 5x + 2x - 10 = x(x - 5) + 2(x - 5)
\][/tex]
5. Factor out the common binomial factor ([tex]\(x - 5\)[/tex]):
[tex]\[
x(x - 5) + 2(x - 5) = (x - 5)(x + 2)
\][/tex]
So, the factored form of [tex]\(x^2 - 3x - 10\)[/tex] is [tex]\((x - 5)(x + 2)\)[/tex].
Next, let's check our factorization using the FOIL (First, Outer, Inner, Last) method:
- First: Multiply the first terms in each binomial:
[tex]\[
x \cdot x = x^2
\][/tex]
- Outer: Multiply the outer terms in each binomial:
[tex]\[
x \cdot 2 = 2x
\][/tex]
- Inner: Multiply the inner terms in each binomial:
[tex]\[
-5 \cdot x = -5x
\][/tex]
- Last: Multiply the last terms in each binomial:
[tex]\[
-5 \cdot 2 = -10
\][/tex]
Combine all the terms:
[tex]\[
x^2 + 2x - 5x - 10 = x^2 - 3x - 10
\][/tex]
After performing the FOIL method, we see that the expanded form of [tex]\((x - 5)(x + 2)\)[/tex] is indeed [tex]\(x^2 - 3x - 10\)[/tex], confirming that our factorization is correct.
Therefore, the trinomial [tex]\(x^2 - 3x - 10\)[/tex] factors completely as:
[tex]\[
\boxed{(x - 5)(x + 2)}
\][/tex]
