Certainly! Let’s go through the process of factoring the given polynomial step by step.
### Problem Statement
We need to factorize the polynomial:
[tex]\[ x^2 - 5x + 6 \][/tex]
And identify which of the given options is the correct factorization.
### Steps to Factorize the Polynomial
1. Identify the polynomial:
[tex]\[ x^2 - 5x + 6 \][/tex]
2. Find two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (-5).
Let's list pairs of factors for 6:
- [tex]\(1 \times 6 = 6\)[/tex]
- [tex]\(2 \times 3 = 6\)[/tex]
We need the pair that adds up to -5:
- [tex]\((1) + (-6) = -5\)[/tex]
- [tex]\((2) + (-3) = -1\)[/tex]
- [tex]\((-2) + (-3) = -5\)[/tex]
The pair that works is [tex]\((-2)\)[/tex] and [tex]\((-3)\)[/tex].
3. Write the polynomial as a product of two binomials:
[tex]\[ x^2 - 5x + 6 = (x - 2)(x - 3) \][/tex]
This matches with option B:
[tex]\[ (x - 2)(x - 3) \][/tex]
### Verification
To ensure our factorization is correct, we can expand the factors:
[tex]\[ (x - 2)(x - 3) \][/tex]
[tex]\[ = x(x - 3) - 2(x - 3) \][/tex]
[tex]\[ = x^2 - 3x - 2x + 6 \][/tex]
[tex]\[ = x^2 - 5x + 6 \][/tex]
The expanded form matches the original polynomial.
### Conclusion
The correct factorization of the polynomial [tex]\( x^2 - 5x + 6 \)[/tex] is:
[tex]\[ (x - 2)(x - 3) \][/tex]
Thus, the correct answer is:
[tex]\[ \text{B. } (x - 2)(x - 3) \][/tex]
