(2x+3y)^5 Binomial Expansion

Jasonbradley

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Jun 16, 2024

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Expanding the Binomial (2x + 3y)^5

The binomial theorem provides a powerful tool for expanding expressions of the form (a + b)^n, where n is a positive integer. In this case, we'll explore the expansion of (2x + 3y)^5.

Understanding the Binomial Theorem

The binomial theorem states:

(a + b)^n = n! / (0! * n!) * a^n * b^0 + n! / (1! * (n-1)!) * a^(n-1) * b^1 + n! / (2! * (n-2)!) * a^(n-2) * b^2 + ... + n! / (n! * 0!) * a^0 * b^n

where "!" denotes the factorial operation (e.g., 5! = 5 * 4 * 3 * 2 * 1).

Applying the Theorem to (2x + 3y)^5

Let's break down the expansion:

1. Identify a and b: In this case, a = 2x and b = 3y.

2. Determine n: Here, n = 5.

3. Apply the formula:

   • Term 1: (5! / (0! * 5!)) * (2x)^5 * (3y)^0 = 32x^5

   • Term 2: (5! / (1! * 4!)) * (2x)^4 * (3y)^1 = 240x^4y

   • Term 3: (5! / (2! * 3!)) * (2x)^3 * (3y)^2 = 720x^3y^2

   • Term 4: (5! / (3! * 2!)) * (2x)^2 * (3y)^3 = 1080x^2y^3

   • Term 5: (5! / (4! * 1!)) * (2x)^1 * (3y)^4 = 810xy^4

   • Term 6: (5! / (5! * 0!)) * (2x)^0 * (3y)^5 = 243y^5

4. Combine the terms:

(2x + 3y)^5 = 32x^5 + 240x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5

Key Points

• Pascal's Triangle: The coefficients in the binomial expansion (32, 240, 720, 1080, 810, 243) can be found in Pascal's Triangle. Each row represents the coefficients for a different power of the binomial.

• Symmetry: Notice that the coefficients are symmetrical. This is a general pattern in binomial expansions.

• Exponents: Observe that the exponents of 'x' decrease from 5 to 0, while the exponents of 'y' increase from 0 to 5.

By applying the binomial theorem, we can efficiently expand expressions like (2x + 3y)^5, revealing the intricate relationships between the terms.