%5E3-formula-class-9.jpeg "(a+b+c)^3 Formula Class 9")
(a + b + c)³ Formula: A Comprehensive Guide for Class 9
The formula (a + b + c)³ is a fundamental concept in algebra, often encountered in Class 9 mathematics. Understanding this formula and its application is crucial for solving various algebraic problems.
What is the (a + b + c)³ Formula?
The expanded form of (a + b + c)³ is:
(a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc
Derivation of the Formula
The formula can be derived using the distributive property and the binomial theorem. Here's a step-by-step breakdown:
-
Square the first binomial: (a + b + c)² = (a + b + c)(a + b + c) = a² + ab + ac + ba + b² + bc + ca + cb + c² = a² + b² + c² + 2ab + 2ac + 2bc
-
Multiply the result by (a + b + c): (a + b + c)³ = (a + b + c)(a² + b² + c² + 2ab + 2ac + 2bc) = a³ + ab² + ac² + 2a²b + 2a²c + 2abc + ba² + b³ + bc² + 2ab² + 2abc + 2b²c + ca² + cb² + c³ + 2abc + 2ac² + 2bc²
-
Combine like terms: = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc
Applications of the Formula
The (a + b + c)³ formula is useful in various algebraic scenarios:
- Expanding expressions: You can use it to expand expressions containing (a + b + c) cubed.
- Factoring expressions: This formula can be used in reverse to factor expressions into the form (a + b + c)³.
- Solving equations: The formula can be applied to solve equations involving cubic expressions.
Example:
Expand the expression (x + 2y + 3z)³:
Using the formula, we get:
(x + 2y + 3z)³ = x³ + (2y)³ + (3z)³ + 3(x²)(2y) + 3(x²)(3z) + 3(x)(2y)² + 3(x)(3z)² + 3(2y)²(3z) + 3(2y)(3z)² + 6(x)(2y)(3z)
Simplifying, we get:
(x + 2y + 3z)³ = x³ + 8y³ + 27z³ + 6x²y + 9x²z + 12xy² + 27xz² + 36y²z + 54yz² + 36xyz
Conclusion
The (a + b + c)³ formula is a valuable tool in algebra, providing a shortcut to expanding and factoring expressions involving cubic terms. By understanding its derivation and applications, you can confidently tackle various algebraic problems in Class 9 and beyond.