## Mastering the (a + b)³ Formula: A Comprehensive Guide to Problem Solving

The formula (a + b)³ is a fundamental concept in algebra, with wide applications in various fields. Understanding this formula and its applications can significantly enhance your problem-solving skills.

### The Formula and its Derivation

The formula (a + b)³ expands to **a³ + 3a²b + 3ab² + b³**. This expansion can be derived using the distributive property of multiplication:

**(a + b)³ = (a + b)(a + b)(a + b)**

First, we multiply the first two factors:

**(a + b)(a + b) = a² + 2ab + b²**

Then, we multiply this result by the remaining factor:

**(a² + 2ab + b²)(a + b) = a³ + 3a²b + 3ab² + b³**

This provides us with the final expanded form of (a + b)³.

### Types of Problems and Solutions

Here are some common types of problems involving the (a + b)³ formula and how to solve them:

**1. Direct Application:**

**Problem:**Expand (2x + 3y)³.**Solution:**Directly apply the formula, substituting 'a' with 2x and 'b' with 3y: (2x + 3y)³ = (2x)³ + 3(2x)²(3y) + 3(2x)(3y)² + (3y)³ = 8x³ + 36x²y + 54xy² + 27y³

**2. Simplification and Evaluation:**

**Problem:**Simplify and evaluate the expression (x + 2)³ - 3x(x + 2)² when x = 1.**Solution:**First, expand using the formula: (x + 2)³ - 3x(x + 2)² = x³ + 6x² + 12x + 8 - 3x(x² + 4x + 4) = x³ + 6x² + 12x + 8 - 3x³ - 12x² - 12x = -2x³ - 6x² + 8 Now, substitute x = 1: -2(1)³ - 6(1)² + 8 = -2 - 6 + 8 = 0

**3. Equation Solving:**

**Problem:**Solve the equation (x + 1)³ - (x - 1)³ = 26.**Solution:**Expand using the formula: (x³ + 3x² + 3x + 1) - (x³ - 3x² + 3x - 1) = 26 Simplify: 6x² + 2 = 26 Solve for x: 6x² = 24 x² = 4 x = ±2

**4. Word Problems:**

**Problem:**The volume of a cube is increasing at a rate of 12 cm³/s. Find the rate at which the side length of the cube is increasing when the side length is 2 cm.**Solution:**Let 's' be the side length of the cube. Then, the volume 'V' is given by V = s³. We are given dV/dt = 12 cm³/s. We need to find ds/dt when s = 2 cm. Differentiating the volume equation with respect to time: dV/dt = 3s² ds/dt Substituting the given values: 12 = 3(2)² ds/dt Solving for ds/dt: ds/dt = 1 cm/s

### Conclusion

The (a + b)³ formula is a powerful tool for simplifying expressions, solving equations, and tackling various problems in algebra and beyond. By understanding the formula's derivation, practicing different problem types, and applying it to real-world scenarios, you can gain a deeper understanding of this fundamental concept and enhance your mathematical abilities.