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(3x-2)^3 Expand

3 min read Jun 16, 2024
(3x-2)^3 Expand

Expanding (3x - 2)^3

In mathematics, expanding an expression like (3x - 2)^3 means writing it out as a sum of terms. We can do this using the binomial theorem or by repeated multiplication.

Using the Binomial Theorem

The binomial theorem states that:

(a + b)^n = a^n + nCa * a^(n-1)b + nC2 * a^(n-2)b^2 + ... + nCn-1 * ab^(n-1) + b^n

where:

  • n is a positive integer representing the power
  • nCk is the binomial coefficient, calculated as n! / (k! * (n-k)!)

Let's apply this to our expression (3x - 2)^3:

  1. Identify a and b: In our case, a = 3x and b = -2.

  2. Apply the formula: (3x - 2)^3 = (3x)^3 + 3C1 * (3x)^2 * (-2) + 3C2 * (3x)^1 * (-2)^2 + (-2)^3

  3. Calculate the binomial coefficients:

  • 3C1 = 3! / (1! * 2!) = 3
  • 3C2 = 3! / (2! * 1!) = 3
  1. Simplify the expression: (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8

Expanding by Repeated Multiplication

We can also expand (3x - 2)^3 by multiplying it out:

  1. First multiplication: (3x - 2)^3 = (3x - 2) * (3x - 2) * (3x - 2) = (9x^2 - 12x + 4) * (3x - 2)

  2. Second multiplication: = 27x^3 - 36x^2 + 12x - 18x^2 + 24x - 8

  3. Combine like terms: = 27x^3 - 54x^2 + 36x - 8

Conclusion

Both methods lead to the same result: (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8

Choosing the method depends on personal preference and the complexity of the expression. The binomial theorem is more efficient for higher powers, while repeated multiplication might be easier to grasp for simpler expressions.

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