(3x-2)^4 Expand

3 min read Jun 16, 2024

Expanding (3x - 2)^4

Expanding the expression (3x - 2)^4 can be done in several ways, but we'll focus on two common methods:

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n. In this case, we have:

(3x - 2)^4 = ⁴C₀(3x)⁴(-2)⁰ + ⁴C₁(3x)³(-2)¹ + ⁴C₂(3x)²(-2)² + ⁴C₃(3x)¹(-2)³ + ⁴C₄(3x)⁰(-2)⁴

Where ⁴Cᵣ represents the binomial coefficient, calculated as ⁴Cᵣ = 4! / (r! * (4-r)!).

Let's break down each term:

⁴C₀(3x)⁴(-2)⁰ = 1 * 81x⁴ * 1 = 81x⁴
⁴C₁(3x)³(-2)¹ = 4 * 27x³ * -2 = -216x³
⁴C₂(3x)²(-2)² = 6 * 9x² * 4 = 216x²
⁴C₃(3x)¹(-2)³ = 4 * 3x * -8 = -96x
⁴C₄(3x)⁰(-2)⁴ = 1 * 1 * 16 = 16

Therefore, the expanded form of (3x - 2)^4 is:

(3x - 2)⁴ = 81x⁴ - 216x³ + 216x² - 96x + 16

We can also expand (3x - 2)^4 by repeatedly multiplying the expression by itself.

(3x - 2)⁴ = (3x - 2)(3x - 2)(3x - 2)(3x - 2)

First, multiply the first two factors:

(3x - 2)(3x - 2) = 9x² - 12x + 4

Then, multiply this result by the third factor:

(9x² - 12x + 4)(3x - 2) = 27x³ - 36x² + 12x - 18x² + 24x - 8 = 27x³ - 54x² + 36x - 8

Finally, multiply this result by the last factor:

(27x³ - 54x² + 36x - 8)(3x - 2) = 81x⁴ - 162x³ + 108x² - 24x - 54x³ + 108x² - 72x + 16 = 81x⁴ - 216x³ + 216x² - 96x + 16

As you can see, both methods lead to the same result:

(3x - 2)⁴ = 81x⁴ - 216x³ + 216x² - 96x + 16

You can choose the method you find most comfortable or convenient for your specific situation.