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g) \( \quad(3 x+2 y)^{3} \)

Ask by Gough Rodriquez. in South Africa
Jan 25,2025

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Tutor-Verified Answer

Answer

\( (3x + 2y)^3 = 27x^3 + 54x^2y + 36xy^2 + 8y^3 \)

Solution

To expand the expression \( (3x + 2y)^3 \), you can use the **Binomial Theorem**, which states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] For \( n = 3 \), the expansion becomes: \[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \] Applying this to \( (3x + 2y)^3 \): \[ (3x + 2y)^3 = (3x)^3 + 3(3x)^2(2y) + 3(3x)(2y)^2 + (2y)^3 \] Now, calculate each term: 1. **First Term:** \[ (3x)^3 = 27x^3 \] 2. **Second Term:** \[ 3(3x)^2(2y) = 3 \times 9x^2 \times 2y = 54x^2y \] 3. **Third Term:** \[ 3(3x)(2y)^2 = 3 \times 3x \times 4y^2 = 36xy^2 \] 4. **Fourth Term:** \[ (2y)^3 = 8y^3 \] **Putting it all together:** \[ (3x + 2y)^3 = 27x^3 + 54x^2y + 36xy^2 + 8y^3 \] **Final Expanded Form:** \[ 27x^3 + 54x^2y + 36xy^2 + 8y^3 \]

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The Deep Dive

To expand \( (3x + 2y)^{3} \), we can use the binomial theorem, which states that \( (a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). Here, \( a = 3x \), \( b = 2y \), and \( n = 3 \). Expanding it gives: \[ (3x + 2y)^{3} = \binom{3}{0}(3x)^{3}(2y)^{0} + \binom{3}{1}(3x)^{2}(2y)^{1} + \binom{3}{2}(3x)^{1}(2y)^{2} + \binom{3}{3}(3x)^{0}(2y)^{3} \] Calculating each term, we find: 1. \( \binom{3}{0}(3x)^{3} = 1 \cdot 27x^{3} = 27x^{3} \) 2. \( \binom{3}{1}(3x)^{2}(2y) = 3 \cdot 9x^{2}(2y) = 54x^{2}y \) 3. \( \binom{3}{2}(3x)(2y)^{2} = 3 \cdot 3x(4y^{2}) = 36xy^{2} \) 4. \( \binom{3}{3}(2y)^{3} = 1 \cdot 8y^{3} = 8y^{3} \) Now combining all the terms together: \[ (3x + 2y)^{3} = 27x^{3} + 54x^{2}y + 36xy^{2} + 8y^{3} \]

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