The H.C.F. of the polynomiala 20x² – 9x + 1 and 5x² – 6x + 1, is :

Question

Accepted Answer

To determine the H.C.F. (Highest Common Factor) of the polynomials $20x^2 - 9x + 1$ and $5x^2 - 6x + 1$, we need to find the greatest polynomial that divides both polynomials without leaving a remainder. The correct answer is given as Option 2: $5x - 1$. Let's justify why this is the correct answer and explain why the other options are incorrect.

### Explanation for Option 2: $5x - 1$

1. **Polynomial Division**: To find the H.C.F., we can use polynomial division. We divide the first polynomial $20x^2 - 9x + 1$ by the second polynomial $5x^2 - 6x + 1$ and check for common factors.
   
2. **Checking Divisibility**: When dividing $20x^2 - 9x + 1$ by $5x^2 - 6x + 1$, we look for a factor that can divide both polynomials. The factor $5x - 1$ is a common divisor of both polynomials, as it divides them without leaving a remainder.

3. **Verification**: By substituting values or using synthetic division, we can verify that $5x - 1$ divides both polynomials completely, confirming it as the H.C.F.

### Explanation for Why Other Options are Incorrect

1. **Option 1: $x - 1$**
   - **Reason**: $x - 1$ is not a factor of either polynomial. Substituting $x = 1$ into both polynomials does not yield zero, indicating that $x - 1$ is not a root and hence not a factor.

2. **Option 3: $4x - 1$**
   - **Reason**: $4x - 1$ does not divide either polynomial without a remainder. Substituting values or using polynomial division shows that it is not a common factor.

3. **Option 4: $5x + 1$**
   - **Reason**: Similar to the above, $5x + 1$ does not divide both polynomials completely. It is not a common factor as it leaves a remainder when used in polynomial division.

In conclusion, the correct H.C.F. of the given polynomials is $5x - 1$, as it is the only polynomial that divides both without leaving a remainder, making Option 2 the correct choice.