(2y-3x)^5 Binomial Theorem

Jasonbradley

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Jun 16, 2024

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Expanding (2y - 3x)^5 with the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. This theorem is extremely useful for expanding expressions with high powers.

Understanding the Binomial Theorem

The binomial theorem states that: (x + y)^n = Σ (n choose k) * x^(n-k) * y^k

where:

• n is the power to which the binomial is raised.
• k is an integer that ranges from 0 to n.
• (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items. It is calculated as n!/(k!(n-k)!).
• Σ represents the sum of the terms from k = 0 to k = n.

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

Let's apply the binomial theorem to expand (2y - 3x)^5.

1. Identify x and y: In this case, x = -3x and y = 2y.
2. Determine n: n = 5.

Now, let's expand the expression using the binomial theorem formula:

(2y - 3x)^5 = Σ (5 choose k) * (-3x)^(5-k) * (2y)^k

Let's calculate the terms for k = 0 to k = 5:

• k = 0: (5 choose 0) * (-3x)^5 * (2y)^0 = 1 * (-243x^5) * 1 = -243x^5
• k = 1: (5 choose 1) * (-3x)^4 * (2y)^1 = 5 * (81x^4) * (2y) = 810x^4y
• k = 2: (5 choose 2) * (-3x)^3 * (2y)^2 = 10 * (-27x^3) * (4y^2) = -1080x^3y^2
• k = 3: (5 choose 3) * (-3x)^2 * (2y)^3 = 10 * (9x^2) * (8y^3) = 720x^2y^3
• k = 4: (5 choose 4) * (-3x)^1 * (2y)^4 = 5 * (-3x) * (16y^4) = -240xy^4
• k = 5: (5 choose 5) * (-3x)^0 * (2y)^5 = 1 * 1 * (32y^5) = 32y^5

Final Result

By adding all the terms, we get the expanded form of (2y - 3x)^5:

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

Key Points

• The binomial theorem is a powerful tool for expanding expressions with high powers.
• The binomial coefficient represents the number of combinations of choosing k items from a set of n items.
• The formula provides a systematic way to calculate each term in the expansion.
• Understanding the binomial theorem can be beneficial in various fields, including mathematics, statistics, and physics.