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Which is the correct expansion of (z +4y)³ using the binomial theorem? ° (z + 4y)² = (3) (z)³(43)° + (³) (²)}² (w)² + (2) (-)³(w)² + (3) (-2°(43)³ ○ (z+4y)² = (4) + (z)²(4y)³¹ + (z)¹ (4y)² + (z)³(49)³ * (z + 4y)² = (3) (43)³¹ (2)° + (³) (4³(z)³¹ + (2) 4)³(2³² + (3) 4°(z)³ O (z+4y)² = (z) (4y) + (z)³¹ (4y)² + (z)² (4y)² + (z)³ (4y)³

Which is the correct expansion of (z +4y)³ using the binomial theorem? ° (z + 4y)² = (3) (z)³(43)° + (³) (²)}² (w)² + (2) (-)³(w)² + (3) (-2°(43)³ ○ (z+4y)² = (4) + (z)²(4y)³¹ + (z)¹ (4y)² + (z)³(49)³ * (z + 4y)² = (3) (43)³¹ (2)° + (³) (4³(z)³¹ + (2) 4)³(2³² + (3) 4°(z)³ O (z+4y)² = (z) (4y) + (z)³¹ (4y)² + (z)² (4y)² + (z)³ (4y)³

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Question:** Which is the correct expansion of \((x + 4y)^3\) using the binomial theorem? 1. \((x + 4y)^3 = \binom{3}{0} (x)^3 (4y)^0 + \binom{3}{1} (x)^2 (4y)^1 + \binom{3}{2} (x)^1 (4y)^2 + \binom{3}{3} (x)^0 (4y)^3\) 2. \((x + 4y)^3 = (4y)^0 + (x)^2 (4y)^1 + (x)^1 (4y)^2 + (x)^0 (4y)^3\) 3. \((x + 4y)^3 = \binom{3}{0} (4y)^3 (x)^0 + \binom{3}{1} (4y)^2 (x)^1 + \binom{3}{2} (4y)^1 (x)^2 + \binom{3}{3} (4y)^0 (x)^3\) 4. \((x + 4y)^3 = (x)^0 (4y)^1 + (x)^1 (4y)^1 + (x)^2 (4y)^2 + (x)^3 (4y)^3\) **Explanation:** The binomial theorem states that \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). Based on this theorem, the correct expansion of \((x + 4y)^3\) involves calculating the binomial coefficients, \(\binom{3}{k}\), and multiplying these by the terms raised to the powers as indicated in option 1. Here, the powers of \(x\) start from 3 and decrease to 0, while the powers of \(4y\) start from 0 and increase to 3.
Transcribed Image Text:**Question:** Which is the correct expansion of \((x + 4y)^3\) using the binomial theorem? 1. \((x + 4y)^3 = \binom{3}{0} (x)^3 (4y)^0 + \binom{3}{1} (x)^2 (4y)^1 + \binom{3}{2} (x)^1 (4y)^2 + \binom{3}{3} (x)^0 (4y)^3\) 2. \((x + 4y)^3 = (4y)^0 + (x)^2 (4y)^1 + (x)^1 (4y)^2 + (x)^0 (4y)^3\) 3. \((x + 4y)^3 = \binom{3}{0} (4y)^3 (x)^0 + \binom{3}{1} (4y)^2 (x)^1 + \binom{3}{2} (4y)^1 (x)^2 + \binom{3}{3} (4y)^0 (x)^3\) 4. \((x + 4y)^3 = (x)^0 (4y)^1 + (x)^1 (4y)^1 + (x)^2 (4y)^2 + (x)^3 (4y)^3\) **Explanation:** The binomial theorem states that \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). Based on this theorem, the correct expansion of \((x + 4y)^3\) involves calculating the binomial coefficients, \(\binom{3}{k}\), and multiplying these by the terms raised to the powers as indicated in option 1. Here, the powers of \(x\) start from 3 and decrease to 0, while the powers of \(4y\) start from 0 and increase to 3.
Expert Solution
Step 1

To evaluate: x+4y3

We know that binomial expansion of x+yn is defined as 

x+yn = n0xny0 +n1xn-1y1 +n2xn-2y2 +...+nnx0yn 

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