%5E3-expand.jpeg "(2x-7)^3 Expand")
Expanding (2x - 7)³
Expanding an expression raised to a power involves multiplying the expression by itself the number of times indicated by the exponent. In this case, we need to multiply (2x - 7) by itself three times:
(2x - 7)³ = (2x - 7)(2x - 7)(2x - 7)
We can solve this by multiplying the expressions in stages:
Step 1: Expand the first two factors.
(2x - 7)(2x - 7) = 4x² - 14x - 14x + 49
Step 2: Simplify the expression.
4x² - 28x + 49
Step 3: Multiply the simplified expression from Step 2 by the remaining factor.
(4x² - 28x + 49)(2x - 7) = 8x³ - 56x² + 98x - 28x² + 196x - 343
Step 4: Combine like terms.
8x³ - 84x² + 294x - 343
Therefore, the expanded form of (2x - 7)³ is 8x³ - 84x² + 294x - 343.
Alternatively, you can use the binomial theorem to expand the expression. The binomial theorem states that:
(a + b)ⁿ = ∑(k=0 to n) (nCk) a^(n-k) b^k
Where nCk is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Applying this to our expression:
(2x - 7)³ = ∑(k=0 to 3) (3Ck) (2x)^(3-k) (-7)^k
This expands to:
(3C0)(2x)³(-7)⁰ + (3C1)(2x)²(-7)¹ + (3C2)(2x)¹(-7)² + (3C3)(2x)⁰(-7)³
Simplifying the binomial coefficients and calculating the powers:
8x³ - 84x² + 294x - 343
This demonstrates that both methods lead to the same result: 8x³ - 84x² + 294x - 343.