What is the HCF of $8(x^5 − x^3 + x) and 28(x^6 + 1)$ ?

Question

Accepted Answer

To determine the HCF (Highest Common Factor) of the expressions $8(x^5 − x^3 + x)$ and $28(x^6 + 1)$, we need to find the common factors in both expressions.

### Step-by-Step Explanation:

1. **Factorization of the Expressions:**

- **Expression 1:** $8(x^5 − x^3 + x)$
     - The expression can be factored by taking out the common factor of 8:
     $$
     8(x^5 − x^3 + x) = 8x(x^4 − x^2 + 1)
     $$

- **Expression 2:** $28(x^6 + 1)$
     - The expression can be factored by taking out the common factor of 28:
     $$
     28(x^6 + 1) = 28(x^6 + 1)
     $$

2. **Identify Common Factors:**

- Both expressions have a numerical factor and a polynomial factor.
   - The numerical factors are 8 and 28. The HCF of 8 and 28 is 4.
   - The polynomial factor common to both expressions is $x^4 − x^2 + 1$.

3. **Calculate the HCF:**

- The HCF of the numerical parts is 4.
   - The polynomial part common to both expressions is $x^4 − x^2 + 1$.

Therefore, the HCF of the two expressions is:
   $$
   4(x^4 − x^2 + 1)
   $$

### Explanation of Options:

- **Option 1: $4(x^4 − x^2 + 1)$**
  - This option correctly represents the HCF as calculated above, combining the numerical HCF (4) with the common polynomial factor $(x^4 − x^2 + 1)$.

- **Option 2: $2(x^4 − x^2 + 1)$**
  - This option is incorrect because it does not account for the correct numerical HCF, which is 4, not 2.

- **Option 3: $(x^4 − x^2 + 1)$**
  - This option is incorrect because it omits the numerical HCF, which is 4. The polynomial part alone is not the complete HCF.

- **Option 4: None of the above**
  - This option is incorrect because Option 1 is indeed the correct HCF.

In conclusion, Option 1 is the correct answer as it accurately represents the HCF of the given expressions.