Find the HCF of `x^2 - 5x + 6 and x^2 - 9`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(x^2 - 5x + 6\) and \(x^2 - 9\), we will follow these steps:

### Step 1: Factor the first polynomial \(x^2 - 5x + 6\)

To factor \(x^2 - 5x + 6\), we need to find two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)).

The numbers that satisfy these conditions are \(-2\) and \(-3\).

Thus, we can factor the polynomial as:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Factor the second polynomial \(x^2 - 9\)

The polynomial \(x^2 - 9\) is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 3\).

So, we can factor it as:
\[ x^2 - 9 = (x - 3)(x + 3) \]

### Step 3: Identify the common factors

Now we have the factored forms of both polynomials:
1. \(x^2 - 5x + 6 = (x - 2)(x - 3)\)
2. \(x^2 - 9 = (x - 3)(x + 3)\)

The common factor in both factorizations is \(x - 3\).

### Step 4: Conclusion

Thus, the HCF of the polynomials \(x^2 - 5x + 6\) and \(x^2 - 9\) is:
\[ \text{HCF} = x - 3 \]

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