Final answer:
Answer: a(x) = (x^2 - 1)(x^3 + x^2 + x + 1). b(x) = (x^3 - 1)(x^3 + 1). To find equivalent expressions for a(x) and b(x), factor the numerators as a difference of squares and simplify by canceling out the (x - 1) terms in the denominators.
Step-by-step explanation:
To find equivalent expressions for a(x) and b(x), we can simplify the given expressions.
For a(x) = x^5 - 1 / (x - 1), we can factor the numerator as a difference of squares: x^5 - 1 = (x^2 - 1)(x^3 + x^2 + x + 1). Then, we can simplify the expression by canceling out the (x - 1) terms in the denominator, resulting in a(x) = (x^2 - 1)(x^3 + x^2 + x + 1).
For b(x) = x^6 - 1 / (x - 1), we can also factor the numerator as a difference of squares: x^6 - 1 = (x^3 - 1)(x^3 + 1). We can simplify the expression by canceling out the (x - 1) terms in the denominator, resulting in b(x) = (x^3 - 1)(x^3 + 1).