The HCF of `(x^4-1)` and `(x^3+x^2+x+1)` is:

Options

A. `(x^2-1)(x^2+1)`
B. `(x^2+1)(x+1)`
C. `(x+1)(x^2-1)`
D. `(x+1)(x^2+1)(x^3+1)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \( f(x) = x^4 - 1 \) and \( g(x) = x^3 + x^2 + x + 1 \), we can follow these steps:

### Step 1: Factor \( f(x) = x^4 - 1 \)
We can recognize that \( x^4 - 1 \) can be factored using the difference of squares:
\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) \]
Next, we can factor \( x^2 - 1 \) further since it is also a difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
Thus, we can write:
\[ f(x) = (x - 1)(x + 1)(x^2 + 1) \]

### Step 2: Factor \( g(x) = x^3 + x^2 + x + 1 \)
We can factor \( g(x) \) by grouping:
\[ g(x) = x^2(x + 1) + 1(x + 1) = (x + 1)(x^2 + 1) \]

### Step 3: Identify the common factors
Now we have:
- \( f(x) = (x - 1)(x + 1)(x^2 + 1) \)
- \( g(x) = (x + 1)(x^2 + 1) \)

The common factors between \( f(x) \) and \( g(x) \) are \( (x + 1) \) and \( (x^2 + 1) \).

### Step 4: Write the HCF
Thus, the HCF of \( f(x) \) and \( g(x) \) is:
\[ \text{HCF} = (x + 1)(x^2 + 1) \]

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

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