(2x+1)^4 Expand

3 min read Jun 16, 2024
(2x+1)^4 Expand

Expanding (2x + 1)^4

Expanding expressions of the form (ax + b)^n, where n is a positive integer, can be a tedious process. However, there are a couple of methods that make it easier.

1. Binomial Theorem

The binomial theorem provides a formula for expanding any binomial raised to a power:

(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k

where:

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

Let's apply this to (2x + 1)^4:

  • a = 2x
  • b = 1
  • n = 4

Expanding using the formula:

(2x + 1)^4 = (4 choose 0) * (2x)^4 * 1^0 + (4 choose 1) * (2x)^3 * 1^1 + (4 choose 2) * (2x)^2 * 1^2 + (4 choose 3) * (2x)^1 * 1^3 + (4 choose 4) * (2x)^0 * 1^4

Calculating the binomial coefficients:

  • (4 choose 0) = 1
  • (4 choose 1) = 4
  • (4 choose 2) = 6
  • (4 choose 3) = 4
  • (4 choose 4) = 1

Substituting the values and simplifying:

(2x + 1)^4 = 1 * 16x^4 * 1 + 4 * 8x^3 * 1 + 6 * 4x^2 * 1 + 4 * 2x * 1 + 1 * 1 * 1

Finally, we get:

(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1

2. Pascal's Triangle

Pascal's Triangle provides a visual representation of the binomial coefficients. Each row represents the coefficients for a different power of a binomial.

To expand (2x + 1)^4, we need the 5th row of Pascal's Triangle:

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1

The coefficients are 1, 4, 6, 4, and 1. Now, we use these coefficients, following the pattern of the binomial theorem:

(2x + 1)^4 = 1(2x)^4 + 4(2x)^31 + 6(2x)^21^2 + 4(2x)1^3 + 11^4**

Simplifying:

(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1

Both methods lead to the same result:

(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1

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