Find HCF of `(x^2 - 3x+ 2) and (x^2- 4x + 3).`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(x^2 - 3x + 2\) and \(x^2 - 4x + 3\), we will follow these steps:

### Step 1: Factorize the Polynomials

**For the first polynomial \(x^2 - 3x + 2\):**

1. We need to express \(x^2 - 3x + 2\) in factored form.
2. We can rewrite it as:
\[ x^2 - 3x + 2 = x^2 - 2x - x + 2 \]
3. Now, we can group the terms:
\[ = x(x - 2) - 1(x - 2) \]
4. Factor out the common term \((x - 2)\):
\[ = (x - 2)(x - 1) \]

**For the second polynomial \(x^2 - 4x + 3\):**

1. We need to express \(x^2 - 4x + 3\) in factored form.
2. We can rewrite it as:
\[ x^2 - 4x + 3 = x^2 - 3x - x + 3 \]
3. Now, we can group the terms:
\[ = x(x - 3) - 1(x - 3) \]
4. Factor out the common term \((x - 3)\):
\[ = (x - 3)(x - 1) \]

### Step 2: Identify the Common Factors

Now we have the factorizations:
- \(x^2 - 3x + 2 = (x - 1)(x - 2)\)
- \(x^2 - 4x + 3 = (x - 1)(x - 3)\)

The common factor between these two factorizations is:
\[ x - 1 \]

### Step 3: Multiply the Common Factors

Since the only common factor we found is \(x - 1\), the HCF of the two polynomials is:
\[ \text{HCF} = x - 1 \]

### Final Answer:
The HCF of \(x^2 - 3x + 2\) and \(x^2 - 4x + 3\) is \(x - 1\).

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