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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.