(2x+3)^4 Binomial Expansion

Jasonbradley

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Jun 16, 2024

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Expanding (2x+3)^4 using the Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions of the form (a + b)^n. It states:

(a + b)^n = Σ (n choose k) a^(n-k) b^k, where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!), and the sum is taken from k = 0 to k = n.

Let's apply this to expand (2x + 3)^4.

1. Identify 'a' and 'b'

• a = 2x
• b = 3

2. Identify 'n'

• n = 4

3. Apply the Binomial Theorem

(2x + 3)^4 = Σ (4 choose k) (2x)^(4-k) (3)^k

4. Expand the Summation

This gives us:

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

5. Calculate the Binomial Coefficients and Simplify

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

Substituting these values and simplifying, we get:

1 * 16x^4 * 1 + 4 * 8x^3 * 3 + 6 * 4x^2 * 9 + 4 * 2x * 27 + 1 * 1 * 81

6. Final Expansion

This simplifies to:

16x^4 + 96x^3 + 216x^2 + 216x + 81

Therefore, the expansion of (2x + 3)^4 is 16x^4 + 96x^3 + 216x^2 + 216x + 81.