%5E3-formula-class-10.jpeg "(a+b)^3 Formula Class 10")
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.