## Fundamental Identities ### Converting a Difference into a Product $\forall b \neq 0$, $ a - b = b \left( \frac{a}{b} - 1 \right) $ It's useful for: - **Sign analysis**. It's often easier to determine the sign of a product than a difference. - **Rate of growth or decay**. The term $\left( \frac{a}{b} - 1 \right)$ represents the relative change from $a$ to $b$. ### Square of a Sum $ (a+b)^2=a^2+2ab+b^2 $ ### Square of a Difference $ (a-b)^2=a^2-2ab+b^2 $ ### Difference of Squares $ a^2-b^2=(a+b)(a-b) $ ## Cubic Identities ### Cube of a Sum $ (a+b)^3=a^3+3a^2b +3ab^2-b^3 $ ### Cube of a Difference $ (a-b)^3=a^3 -3a^2b+3ab^2-b^3 $ ### Sum of Cubes $ a^3+b^3=(a+b)(a^2-ab+b^2) $ ### Difference of Cubes $ a^3-b^3=(a-b)(a^2+ab+b^2) $