Check Your Understanding - Question 1 of 3

Start by factoring [tex]$x^2 - 5x + 6$[/tex].

Which factorization is correct?

A. [tex]$(x - 6)(x + 1)$[/tex]
B. [tex][tex]$(x - 2)(x - 3)$[/tex][/tex]
C. [tex]$(x - 2)(x + 3)$[/tex]
D. [tex]$(x + 6)(x - 1)$[/tex]



Answer :

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]