HCF of `(1+x+x^(2))` and `1-x^(3)`, is

Options

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

Answer

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

### Step 1: Factorize \(1 - x^3\)
We can use the identity for the difference of cubes, which states:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In our case, we can rewrite \(1 - x^3\) as:
\[ 1^3 - x^3 = (1 - x)(1^2 + 1 \cdot x + x^2) = (1 - x)(1 + x + x^2) \]

### Step 2: Identify the common factors
Now we have:
\[ 1 - x^3 = (1 - x)(1 + x + x^2) \]
The first polynomial \(1 + x + x^2\) is already in its simplest form.

### Step 3: Determine the HCF
From the factorization, we can see that \(1 + x + x^2\) is a common factor in both polynomials:
- \(1 + x + x^2\) is present in \(1 - x^3\)
- \(1 + x + x^2\) is also the first polynomial.

Thus, the HCF of \(1 + x + x^2\) and \(1 - x^3\) is:
\[ \text{HCF} = 1 + x + x^2 \]

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