%5E3.jpeg "(2x+7)^3")
Expanding (2x+7)³
In mathematics, expanding expressions often involves applying the distributive property and simplifying the result. Let's break down how to expand (2x+7)³.
Understanding the Expression
The expression (2x+7)³ represents the product of (2x+7) multiplied by itself three times: (2x+7) * (2x+7) * (2x+7)
Expanding Using the Distributive Property
There are two main approaches to expanding this:
1. Step-by-Step Expansion:
- Step 1: Expand the first two factors: (2x+7) * (2x+7) = 4x² + 14x + 14x + 49 = 4x² + 28x + 49
- Step 2: Now multiply the result by the remaining (2x+7): (4x² + 28x + 49) * (2x+7) = 8x³ + 56x² + 98x + 56x² + 392x + 343
- Step 3: Combine like terms: 8x³ + 112x² + 490x + 343
2. Using the Binomial Theorem:
The binomial theorem provides a formula to expand any expression of the form (a+b)ⁿ. In this case, a=2x, b=7, and n=3.
- The Formula: (a + b)³ = a³ + 3a²b + 3ab² + b³
- Applying the Formula: (2x + 7)³ = (2x)³ + 3(2x)²(7) + 3(2x)(7)² + 7³
- Simplifying: 8x³ + 84x² + 294x + 343
Conclusion
Both methods lead to the same result: (2x+7)³ = 8x³ + 112x² + 490x + 343. Expanding expressions is a fundamental skill in algebra, and understanding these techniques will help you solve various problems involving polynomials.