The H.C.F. of expression `(x+1)(x-1)^2` and `(x+1)^2 (x-1)` is

Options

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

Answer

To find the H.C.F. (Highest Common Factor) of the expressions \((x + 1)(x - 1)^2\) and \((x + 1)^2(x - 1)\), we can follow these steps:

### Step 1: Factor the expressions
The first expression is:
\[ (x + 1)(x - 1)^2 \]
The second expression is:
\[ (x + 1)^2(x - 1) \]

### Step 2: Identify the common factors
Now we will look for the common factors in both expressions.

- The first expression \((x + 1)(x - 1)^2\) can be broken down into:
- One factor of \((x + 1)\)
- Two factors of \((x - 1)\)

- The second expression \((x + 1)^2(x - 1)\) can be broken down into:
- Two factors of \((x + 1)\)
- One factor of \((x - 1)\)

### Step 3: Determine the minimum power of each common factor
Now we will find the minimum power of each common factor present in both expressions:

- For \((x + 1)\):
- In the first expression: 1
- In the second expression: 2
- Minimum power: 1

- For \((x - 1)\):
- In the first expression: 2
- In the second expression: 1
- Minimum power: 1

### Step 4: Write the H.C.F.
Now we can write the H.C.F. using the common factors with their minimum powers:
\[ \text{H.C.F.} = (x + 1)^{\text{min}(1, 2)}(x - 1)^{\text{min}(2, 1)} = (x + 1)^1(x - 1)^1 = (x + 1)(x - 1) \]

### Final Answer:
Thus, the H.C.F. of the expressions \((x + 1)(x - 1)^2\) and \((x + 1)^2(x - 1)\) is:
\[ \boxed{(x + 1)(x - 1)} \]