%5E3.jpeg "(2x-7)^3")
Expanding (2x-7)^3
The expression (2x-7)^3 represents the cube of the binomial (2x-7). This means we're multiplying the binomial by itself three times:
(2x-7)^3 = (2x-7)(2x-7)(2x-7)
To expand this expression, we can use the following methods:
1. Repeated Multiplication:
- Step 1: Multiply the first two binomials: (2x-7)(2x-7) = 4x² - 14x - 14x + 49 = 4x² - 28x + 49
- Step 2: Multiply the result from step 1 by the third binomial: (4x² - 28x + 49)(2x-7) = 8x³ - 56x² + 98x - 28x² + 196x - 343
- Step 3: Combine like terms: 8x³ - 84x² + 294x - 343
2. Using the Binomial Theorem:
The binomial theorem provides a formula to expand any binomial raised to a power. The formula for (a + b)^n is:
(a + b)^n = ∑(n choose k) a^(n-k) b^k where k goes from 0 to n
- Step 1: Identify 'a' and 'b' in our binomial: a = 2x and b = -7
- Step 2: Apply the formula for n=3: (2x - 7)^3 = (3 choose 0)(2x)³(-7)⁰ + (3 choose 1)(2x)²(-7)¹ + (3 choose 2)(2x)¹(-7)² + (3 choose 3)(2x)⁰(-7)³
- Step 3: Simplify: (2x - 7)^3 = 8x³ - 84x² + 294x - 343
Conclusion:
Both methods result in the same expanded form of (2x-7)^3: 8x³ - 84x² + 294x - 343. The choice of method depends on personal preference and the complexity of the expression. Remember to carefully apply the distributive property or binomial theorem to ensure accurate expansion.