Final answer:
The HCF of the algebraic expressions (a³ - 1) and (a⁴ + a² + 1) is 1, as the expressions share no common factors other than the multiplicative identity.
Step-by-step explanation:
The question asks us to find the Highest Common Factor (HCF) of two algebraic expressions, (a³ - 1) and (a⁴ + a² + 1).
To find the HCF of two expressions, we can employ various algebraic techniques such as factorization. The expression (a³ - 1) is a difference of cubes, which can be factored into (a - 1)(a² + a + 1).
Looking at the second expression, (a⁴ + a² + 1), it might not be immediately obvious how it factors if at all.
However, if we add and subtract a³ in the expression, we could write it as (a⁴ + a³ + a² - a³ + a + 1) which allows us to group terms and see a pattern.
Unfortunately, on further analysis, it becomes clear that the second expression is prime with respect to real coefficients and thus cannot be factored in a way that reveals a common factor with (a³ - 1).
Hence, the HCF of (a³ - 1) and (a⁴ + a² + 1) is 1, since they share no common factors other than the multiplicative identity.