To factor the trinomial [tex]\(x^2 - 9x + 20\)[/tex], we need to find two binomials that multiply together to give us the original trinomial. Let's break down the steps to find these factors:
1. Identify the coefficient of [tex]\(x^2\)[/tex], the coefficient of [tex]\(x\)[/tex], and the constant term:
- The coefficient of [tex]\(x^2\)[/tex] is [tex]\(1\)[/tex] (since we have [tex]\(x^2\)[/tex]).
- The coefficient of [tex]\(x\)[/tex] is [tex]\(-9\)[/tex].
- The constant term is [tex]\(20\)[/tex].
2. Look for two numbers that multiply to the constant term (20) and add up to the coefficient of [tex]\(x\)[/tex] ([tex]\(-9\)[/tex]):
- We need two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(a \cdot b = 20\)[/tex] (the constant term) and [tex]\(a + b = -9\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
3. Find the appropriate pairs of numbers:
- By examining factors of 20, we find the pairs [tex]\((1, 20), (2, 10), (4, 5), ( -1, -20), (-2, -10), (-4, -5)\)[/tex].
- We need to select the pair that adds up to [tex]\(-9\)[/tex]. Upon inspection, we see that the pair [tex]\((-4, -5)\)[/tex] satisfies the condition, as [tex]\((-4) + (-5) = -9\)[/tex].
4. Write the trinomial as a product of two binomials:
- Since [tex]\((-4) \cdot (-5) = 20\)[/tex] and [tex]\((-4) + (-5) = -9\)[/tex], we can factor the trinomial as [tex]\((x - 5)(x - 4)\)[/tex].
Thus, the trinomial [tex]\(x^2 - 9x + 20\)[/tex] factors into:
[tex]\[
(x - 5)(x - 4)
\][/tex]
Therefore, the factors of the trinomial are [tex]\((\boxed{x - 5})\boxed{(x - 4)}\)[/tex].
