%5E3.jpeg "(4n-5)^3")
Expanding (4n - 5)³
The expression (4n - 5)³ represents the cube of the binomial (4n - 5). To expand this, we can use the following methods:
1. Direct Multiplication
This method involves multiplying the binomial by itself three times:
(4n - 5)³ = (4n - 5) * (4n - 5) * (4n - 5)
First, we multiply the first two binomials:
(4n - 5) * (4n - 5) = 16n² - 20n - 20n + 25 = 16n² - 40n + 25
Then, we multiply the result by the remaining binomial:
(16n² - 40n + 25) * (4n - 5) = 64n³ - 160n² + 100n - 80n² + 200n - 125
Finally, we combine like terms:
** (4n - 5)³ = 64n³ - 240n² + 300n - 125 **
2. Binomial Theorem
The Binomial Theorem provides a general formula for expanding any binomial raised to a power. For (4n - 5)³, the formula can be applied as follows:
(4n - 5)³ = ³C₀(4n)³(-5)⁰ + ³C₁(4n)²(-5)¹ + ³C₂(4n)¹(-5)² + ³C₃(4n)⁰(-5)³
where ³C<sub>k</sub> represents the binomial coefficient, which can be calculated as:
³C<sub>k</sub> = 3! / (k! * (3 - k)!)
Applying the formula, we get:
(4n - 5)³ = (1)(64n³) + (3)(16n²)(-5) + (3)(4n)(25) + (1)(1)(-125)
Simplifying the expression, we arrive at the same result:
** (4n - 5)³ = 64n³ - 240n² + 300n - 125 **
Conclusion
Both methods lead to the same expanded form of (4n - 5)³, which is 64n³ - 240n² + 300n - 125. The choice of method depends on personal preference and the complexity of the expression. The Binomial Theorem might be preferred for higher powers, as it offers a more systematic approach.