The HCF of polynomials `x^(3)-1 and x^(2)-1` is __________.

Options

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

Answer

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

### Step 1: Factor the polynomials
First, we will factor both polynomials using the known formulas.

1. **Factor \( x^3 - 1 \)**:
The expression \( x^3 - 1 \) can be factored using the formula for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \( a = x \) and \( b = 1 \):
\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \]

2. **Factor \( x^2 - 1 \)**:
The expression \( x^2 - 1 \) can be factored using the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Here, \( a = x \) and \( b = 1 \):
\[ x^2 - 1 = (x - 1)(x + 1) \]

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

The common factor in both expressions is \( (x - 1) \).

### Step 3: Determine the HCF
The HCF of the two polynomials is the highest common factor that we identified:
\[ \text{HCF} = x - 1 \]

### Final Answer
The HCF of the polynomials \( x^3 - 1 \) and \( x^2 - 1 \) is \( \boxed{x - 1} \).

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