# Solving Expressions: Factor a^2-b^2 and a^3-b^3

• Gus_Chiggins
In summary, for part 1, the expressions a^2-b^2 and a^3-b^3 can be factored as (a+b)(a-b) and (a-b)(a^2+ab+b^2), respectively. For part 2, without using the product rule or chain rule, formulas for g'(a), k'(a), and p'(a) can be found in terms of a, f(a), and f'(a) by using the limit definition of the derivative. This approach may be helpful in solving the problem in part 1 as well.
Gus_Chiggins
Part 1:Factor the expressions a^2-b^2 and a^3-b^3

For part 1, i worked it out to be:
(a+b)(a-b) and (a-b)(a^2+ab+b^2)

Part 2: Suppose f(x) is a function that has a derivative. Let g(x)=x^2f(x), k(x)=$$\sqrt{f(x)}$$, and p(x)=^3$$\sqrt{f(x)}$$. Without using the product rule and w/o using the chain rule, find formulas for g'(a), k'(a), and p'(a) in terms of a, f(a) and f'(a)

Any help would be great for part 2. I really don't know how to tackle this problem. Thanks!

There was no restriction on using the limit definition of the derivative. I think that might be where they're trying to steer you, especially with the problem in part 1.

## 1. What is the first step in factoring a^2-b^2?

The first step in factoring a^2-b^2 is to identify the difference of squares pattern, which is a^2-b^2 = (a+b)(a-b).

## 2. Can a^2-b^2 be factored further?

No, a^2-b^2 is in its simplest form and cannot be factored any further.

## 3. How is factoring a^3-b^3 different from factoring a^2-b^2?

The main difference is that a^3-b^3 is a difference of cubes, which can be factored into (a-b)(a^2+ab+b^2).

## 4. What happens if there are coefficients in front of a^2 or b^2 in the expression a^2-b^2?

If there are coefficients, you can factor them out first and then follow the same steps as factoring a^2-b^2 or a^3-b^3.

## 5. Can you use factoring to solve equations involving a^2-b^2 or a^3-b^3?

Yes, factoring can be used to solve equations involving a^2-b^2 or a^3-b^3 by setting each factor equal to zero and solving for the variables.

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