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Chapter 14: Problem 11

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(3 x+y)^{3}$$

Short Answer

Expert verified
The expanded form of \( (3x+y)^3 \) is \( 27x^3 + 27x^2y + 9xy^2 + y^3 \)

Step by step solution

01

Understand Binomial Theorem

The Binomial Theorem states that for any numbers a and b, and any natural number n, \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}\) where \({n \choose k}\) denotes the combination of n and k.
02

Apply the theorem to the given problem

In this case, a is '3x', b is 'y', and n is 3. By substituting these values into the formula, we get the expanded form. So, \((3x+y)^3 = \sum_{k=0}^{3} {3 \choose k} (3x)^{3-k} (y)^{k}\).
03

Calculate each term in the sum

Calculate each term in the sum separately - \({3 \choose 0}(3x)^3(y)^0 + {3 \choose 1}(3x)^2(y)^1 + {3 \choose 2}(3x)^1(y)^2 + {3 \choose 3}(3x)^0(y)^3 = 27x^3 + 27x^2y + 9xy^2 + y^3\)
04

Write down the final result

Combine all terms to get the final result.

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