## Understanding the (a + b)³ Formula: A Comprehensive Guide for Class 10

The formula (a + b)³ is a fundamental concept in algebra, particularly relevant for students in Class 10. This formula allows us to expand the cube of a binomial expression, making it easier to simplify and solve equations.

### What is the (a + b)³ Formula?

The formula states that:

**(a + b)³ = a³ + 3a²b + 3ab² + b³**

This formula can be derived using the distributive property of multiplication.

### Derivation of the Formula:

**Expand the expression:**(a + b)³ = (a + b)(a + b)(a + b)**Multiply the first two terms:**(a + b)(a + b) = a² + 2ab + b²**Multiply the result by (a + b):**(a² + 2ab + b²)(a + b) = a³ + 3a²b + 3ab² + b³

### Applying the Formula:

The (a + b)³ formula is used in various mathematical applications, including:

**Simplifying algebraic expressions:**By using the formula, you can expand and simplify complex expressions involving cubes of binomials.**Solving equations:**The formula can be applied to solve equations involving cubes of binomials.**Factoring polynomials:**The formula can be used to factorize cubic expressions.

### Examples:

**Example 1:** Expand (x + 2)³

Using the formula: (x + 2)³ = x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8

**Example 2:** Solve the equation: (x + 1)³ = 8

Expanding the left side using the formula: x³ + 3x² + 3x + 1 = 8 x³ + 3x² + 3x - 7 = 0 Now, we can solve this cubic equation to find the value of x.

### Key Points to Remember:

- The (a + b)³ formula is a powerful tool for simplifying and solving algebraic expressions.
- Practice using the formula in various examples to gain confidence.
- Understand the derivation of the formula to grasp its meaning and application.

By understanding the (a + b)³ formula, Class 10 students can strengthen their algebraic foundation and tackle more complex mathematical problems effectively.