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(x+4)^3 Solve

3 min read Jun 16, 2024
(x+4)^3 Solve

Expanding (x + 4)^3

Expanding (x + 4)^3 means multiplying (x + 4) by itself three times:

(x + 4)^3 = (x + 4)(x + 4)(x + 4)

There are two main ways to solve this:

1. Using the Binomial Theorem:

The Binomial Theorem provides a formula to expand expressions of the form (x + y)^n. It states:

(x + y)^n = Σ (n choose k) * x^(n-k) * y^k, where:

  • n is the power
  • k ranges from 0 to n
  • (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!)

Applying this to our case (x + 4)^3:

  • n = 3
  • k = 0, 1, 2, 3

Let's expand:

  • k = 0: (3 choose 0) * x^(3-0) * 4^0 = 1 * x^3 * 1 = x^3
  • k = 1: (3 choose 1) * x^(3-1) * 4^1 = 3 * x^2 * 4 = 12x^2
  • k = 2: (3 choose 2) * x^(3-2) * 4^2 = 3 * x^1 * 16 = 48x
  • k = 3: (3 choose 3) * x^(3-3) * 4^3 = 1 * x^0 * 64 = 64

Adding all the terms together:

(x + 4)^3 = x^3 + 12x^2 + 48x + 64

2. Expanding step by step:

  • Step 1: Expand (x + 4)(x + 4)

    • (x + 4)(x + 4) = x(x + 4) + 4(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16

  • Step 2: Multiply the result by (x + 4)

    • (x^2 + 8x + 16)(x + 4) = x(x^2 + 8x + 16) + 4(x^2 + 8x + 16)
    • = x^3 + 8x^2 + 16x + 4x^2 + 32x + 64
    • = x^3 + 12x^2 + 48x + 64

Both methods arrive at the same answer: (x + 4)^3 = x^3 + 12x^2 + 48x + 64

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