Find HCF and LCM in Simplified form using Factorization method. $$ x^{6}-1, x^{4}+x^{2}+1 $$

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To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the polynomials $$ x^{6}-1 $$ and $$ x^{4}+x^{2}+1 $$ using the factorization method, we will follow these steps:

### Step 1: Factorize $$ x^{6}-1 $$

The expression $$ x^{6}-1 $$ can be factored using the difference of squares and the difference of cubes:

$$
x^{6}-1 = (x^{3}-1)(x^{3}+1)
$$

Next, we can factor $$ x^{3}-1 $$ and $$ x^{3}+1 $$:

$$
x^{3}-1 = (x-1)(x^{2}+x+1)
$$
$$
x^{3}+1 = (x+1)(x^{2}-x+1)
$$

Thus, the complete factorization of $$ x^{6}-1 $$ is:

$$
x^{6}-1 = (x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)
$$

### Step 2: Factorize $$ x^{4}+x^{2}+1 $$

The expression $$ x^{4}+x^{2}+1 $$ can be treated as a quadratic in terms of $$ x^{2} $$:

Let $$ y = x^{2} $$, then we have:

$$
y^{2} + y + 1
$$

To factor $$ y^{2} + y + 1 $$, we can use the quadratic formula:

$$
y = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}
$$

This means $$ y^{2} + y + 1 $$ does not factor over the reals, but we can express it in terms of its roots:

$$
x^{4}+x^{2}+1 = (x^{2} - \frac{-1 + i\sqrt{3}}{2})(x^{2} - \frac{-1 - i\sqrt{3}}{2})
$$

However, for our purposes, we can keep it as $$ x^{4}+x^{2}+1 $$ since it does not share any common factors with $$ x^{6}-1 $$.

### Step 3: Find HCF and LCM

#### HCF

The HCF is the product of the common factors. Since $$ x^{4}+x^{2}+1 $$ does not share any factors with $$ x^{6}-1 $$, we have:

$$
\text{HCF}(x^{6}-1, x^{4}+x^{2}+1) = 1
$$

#### LCM

The LCM is given by the product of the two polynomials divided by their HCF:

$$
\text{LCM}(x^{6}-1, x^{4}+x^{2}+1) = \frac{(x^{6}-1)(x^{4}+x^{2}+1)}{\text{HCF}(x^{6}-1, x^{4}+x^{2}+1)}
$$

Since the HCF is 1, we have:

$$
\text{LCM}(x^{6}-1, x^{4}+x^{2}+1) = (x^{6}-1)(x^{4}+x^{2}+1)
$$

### Final Results

- **HCF**: $$ 1 $$
- **LCM**: $$ (x^{6}-1)(x^{4}+x^{2}+1) $$

This concludes the factorization and calculation of HCF and LCM for the given polynomials.
The HCF of $$ x^{6}-1 $$ and $$ x^{4}+x^{2}+1 $$ is 1, and the LCM is $$ (x^{6}-1)(x^{4}+x^{2}+1) $$.

Find HCF and LCM in Simplified form using Factorization method. \[ x^{6}-1, x^{4}+x^{2}+1 \]

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