The HCF of `x^(5) - 1` and `x^(4) + 2x^(5) - 2x - 1` is :

Options

A. `x^(2) + 1`
B. `x^(2) - 1`
C. `x + 1`
D. `x - 2

Answer

To find the HCF (Highest Common Factor) of the polynomials \( x^5 - 1 \) and \( x^4 + 2x^5 - 2x - 1 \), we can use the polynomial division method. Here is a step-by-step solution:

### Step 1: Factor \( x^5 - 1 \)
The polynomial \( x^5 - 1 \) can be factored using the difference of cubes:
\[ x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1) \]

### Step 2: Substitute \( x = 1 \) into the second polynomial
Next, we will evaluate the second polynomial \( x^4 + 2x^5 - 2x - 1 \) at \( x = 1 \):
\[ 1^4 + 2(1^5) - 2(1) - 1 = 1 + 2 - 2 - 1 = 0 \]
Since the value is 0, \( x - 1 \) is a factor of \( x^4 + 2x^5 - 2x - 1 \).

### Step 3: Perform polynomial long division
Now we will divide \( x^4 + 2x^5 - 2x - 1 \) by \( x - 1 \) to find the other factor.

1. Divide the leading term of the dividend \( 2x^5 \) by the leading term of the divisor \( x \) to get \( 2x^4 \).
2. Multiply \( 2x^4 \) by \( x - 1 \) to get \( 2x^5 - 2x^4 \).
3. Subtract this from the original polynomial:
\[ (x^4 + 2x^5 - 2x - 1) - (2x^5 - 2x^4) = 3x^4 - 2x - 1 \]

4. Now divide \( 3x^4 \) by \( x \) to get \( 3x^3 \).
5. Multiply \( 3x^3 \) by \( x - 1 \) to get \( 3x^4 - 3x^3 \).
6. Subtract:
\[ (3x^4 - 2x - 1) - (3x^4 - 3x^3) = 3x^3 - 2x - 1 \]

7. Continue this process until you reach a remainder or find the complete factorization.

### Step 4: Determine the HCF
Since we found that \( x - 1 \) is a factor of both polynomials, the HCF of \( x^5 - 1 \) and \( x^4 + 2x^5 - 2x - 1 \) is:
\[ \text{HCF} = x - 1 \]

### Final Answer
The HCF of \( x^5 - 1 \) and \( x^4 + 2x^5 - 2x - 1 \) is \( x - 1 \).

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