HCF of `(x+1)"and"(x^(2)-x+1)` is

Options

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

Answer

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

### Step 1: Identify the expressions
We have two expressions:
1. \(A = x + 1\)
2. \(B = x^2 - x + 1\)

### Step 2: Check if \(A\) divides \(B\)
To find the HCF, we need to check if \(A\) divides \(B\) without leaving a remainder. We can do this by performing polynomial long division.

#### Polynomial Long Division:
- Divide \(B\) by \(A\):
\[ \text{Divide } (x^2 - x + 1) \text{ by } (x + 1) \]

1. The leading term of \(B\) is \(x^2\) and the leading term of \(A\) is \(x\). Divide \(x^2\) by \(x\) to get \(x\).
2. Multiply \(x\) by \(A\):
\[ x(x + 1) = x^2 + x \]
3. Subtract this from \(B\):
\[ (x^2 - x + 1) - (x^2 + x) = -2x + 1 \]

### Step 3: Check the remainder
Now we have a new expression:
\[ R = -2x + 1 \]
Next, we need to check if \(A\) divides \(R\).

### Step 4: Check if \(A\) divides \(R\)
Now we will check if \(x + 1\) divides \(-2x + 1\).

1. Divide \(-2x + 1\) by \(x + 1\):
- The leading term of \(-2x\) divided by \(x\) gives \(-2\).
- Multiply \(-2\) by \(x + 1\):
\[ -2(x + 1) = -2x - 2 \]
- Subtract this from \(-2x + 1\):
\[ (-2x + 1) - (-2x - 2) = 3 \]

### Step 5: Final remainder
Now we have:
\[ R = 3 \]
Since \(3\) is a constant and does not contain \(x\), we check if \(x + 1\) divides \(3\). The only common factor between \(x + 1\) and \(3\) is \(1\).

### Conclusion
Since the only common factor we found is \(1\), the HCF of \(x + 1\) and \(x^2 - x + 1\) is:
\[ \text{HCF} = 1 \]

### Answer
The correct option is \(d\) (1).

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