1. Let's start by stating the problem: We want to understand and apply common algebraic identities. 2. The most common algebraic identities are: - 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: $$ a^2 - b^2 = (a-b)(a+b) $$ - 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: $$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $$ 3. These identities help simplify expressions and solve equations efficiently. 4. For example, to expand $$ (x+3)^2 $$, use the square of a sum: $$ (x+3)^2 = x^2 + 2 \cdot x \cdot 3 + 3^2 = x^2 + 6x + 9 $$ 5. To factor $$ x^2 - 9 $$, use the difference of squares: $$ x^2 - 9 = (x-3)(x+3) $$ 6. To expand $$ (2x - 5)^3 $$, use the cube of a difference: $$ (2x - 5)^3 = (2x)^3 - 3(2x)^2 \cdot 5 + 3(2x) \cdot 5^2 - 5^3 = 8x^3 - 60x^2 + 150x - 125 $$ 7. To factor $$ a^3 + b^3 $$, use the sum of cubes identity: $$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$ These identities are fundamental tools in algebra for simplifying and factoring expressions.