Simplify (b^3)(ab-1)((ab)^2+ab+1)
Hint: The last two factors sure look like the difference of two cubes to me. Have you tried that?
To simplify the given expression, we can look for any patterns or common factors that may allow us to simplify.
The hint suggests that the last two factors, (ab-1) and ((ab)^2 + ab + 1), resemble the difference of two cubes. Let's investigate further.
The difference of two cubes can be expressed as:
a^3 - b^3 = (a - b)(a^2 + ab + b^2).
In this case, we have (ab) as a common factor in both (ab-1) and ((ab)^2 + ab + 1). To express them as the difference of two cubes, we need a^3 - b^3. Therefore, we need to rewrite (ab-1) and ((ab)^2 + ab + 1) as a^3 - b^3.
To do this, we can treat ab as our "a" term and 1 as our "b" term for (ab-1). Using the difference of two cubes formula, we have:
(ab)^3 - 1^3 = [(ab) - 1][(ab)^2 + (ab)(1) + 1].
Now, let's examine ((ab)^2 + ab + 1) as a^3 - b^3. We can treat (ab)^2 as our "a" term and ab as our "b" term:
(ab)^3 - (ab)^3 = [(ab)^2 - (ab)][(ab)^2 + (ab)(ab) + (ab)^2].
Now, we can substitute these expressions back into the original equation:
(b^3)(ab-1)((ab)^2 + ab + 1) = (b^3)[(ab)^3 - 1^3][(ab)^3 - (ab)^3].
Simplifying further, we have:
= (b^3)[(ab-1)(ab^2 + ab + 1)].
And that is the simplified form of the given expression.