Find the HCF of the polynomial `30(x^(2)-3x+2) and 50 (x^(2)-2x +1)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \(30(x^2 - 3x + 2)\) and \(50(x^2 - 2x + 1)\), we can follow these steps:

### Step 1: Factor the polynomials

1. **Factor \(30(x^2 - 3x + 2)\)**:
- First, factor the quadratic \(x^2 - 3x + 2\).
- We look for two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)).
- The numbers \(-1\) and \(-2\) satisfy these conditions.
- Therefore, \(x^2 - 3x + 2 = (x - 1)(x - 2)\).
- Now, we can write:
\[ 30(x^2 - 3x + 2) = 30(x - 1)(x - 2) \]

2. **Factor \(50(x^2 - 2x + 1)\)**:
- Next, factor the quadratic \(x^2 - 2x + 1\).
- This is a perfect square trinomial, which can be factored as:
\[ x^2 - 2x + 1 = (x - 1)^2 \]
- Therefore, we can write:
\[ 50(x^2 - 2x + 1) = 50(x - 1)^2 \]

### Step 2: Identify common factors

Now we have:
- \(f(x) = 30(x - 1)(x - 2)\)
- \(g(x) = 50(x - 1)^2\)

### Step 3: Find the HCF

1. **Identify the common factors**:
- The common numerical factor from \(30\) and \(50\) is \(10\).
- The common polynomial factor is \((x - 1)\).
- The minimum power of \((x - 1)\) in both polynomials is \(1\) (since \(f(x)\) has \((x - 1)^1\) and \(g(x)\) has \((x - 1)^2\)).

2. **Combine these to find the HCF**:
- Therefore, the HCF of the two polynomials is:
\[ HCF = 10(x - 1) \]

### Final Answer:
The HCF of the polynomials \(30(x^2 - 3x + 2)\) and \(50(x^2 - 2x + 1)\) is \(10(x - 1)\).

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