Problem 341 For the following exercises, use... [FREE SOLUTION]

Chapter 11: Problem 341

For the following exercises, use the Binomial Theorem to expand each binomial. $$\left(\frac{1}{x}+3 y\right)^{5}$$

Short Answer

Expert verified

The expanded form is $ \frac{1}{x^5} + \frac{15y}{x^4} + \frac{90y^2}{x^3} + \frac{270y^3}{x^2} + \frac{405y^4}{x} + 243y^5 $.

Step by step solution

Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form $(a + b)^n$. It states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$. In this expression, $ \binom{n}{k} $ is a binomial coefficient calculated as $ \frac{n!}{k!(n-k)!} $.

Identify Parameters

In the expression $ \left( \frac{1}{x} + 3y\right)^5 $, identify $ a = \frac{1}{x} $, $ b = 3y $, and $ n = 5 $. These will be substituted into the binomial expansion formula.

Calculate Each Term Using the Binomial Theorem

Calculate each term using the formula $ T_k = \binom{n}{k} a^{n-k} b^k $ for $ k = 0 $ to $ n = 5 $:- For $ k = 0 $, $ T_0 = \binom{5}{0} \left(\frac{1}{x}\right)^5 (3y)^0 = 1 \cdot \frac{1}{x^5} \cdot 1 = \frac{1}{x^5} $.- For $ k = 1 $, $ T_1 = \binom{5}{1} \left(\frac{1}{x}\right)^4 (3y)^1 = 5 \cdot \frac{1}{x^4} \cdot 3y = \frac{15y}{x^4} $.- For $ k = 2 $, $ T_2 = \binom{5}{2} \left(\frac{1}{x}\right)^3 (3y)^2 = 10 \cdot \frac{1}{x^3} \cdot 9y^2 = \frac{90y^2}{x^3} $.- For $ k = 3 $, $ T_3 = \binom{5}{3} \left(\frac{1}{x}\right)^2 (3y)^3 = 10 \cdot \frac{1}{x^2} \cdot 27y^3 = \frac{270y^3}{x^2} $.- For $ k = 4 $, $ T_4 = \binom{5}{4} \left(\frac{1}{x}\right)^1 (3y)^4 = 5 \cdot \frac{1}{x} \cdot 81y^4 = \frac{405y^4}{x} $.- For $ k = 5 $, $ T_5 = \binom{5}{5} \left(\frac{1}{x}\right)^0 (3y)^5 = 1 \cdot 1 \cdot 243y^5 = 243y^5 $.

Write the Expanded Form

Combine the terms from Step 3 to write the expanded form of the binomial:\[ \left( \frac{1}{x} + 3y \right)^5 = \frac{1}{x^5} + \frac{15y}{x^4} + \frac{90y^2}{x^3} + \frac{270y^3}{x^2} + \frac{405y^4}{x} + 243y^5 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion

Expanding a binomial expression using the Binomial Theorem is a powerful tool in algebra. The Binomial Theorem formula \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]gives us a way to express any binomial raised to a power as a sum of terms. Each term in the expansion is constructed using specific components:

The base expressions, $ a $ and $ b $, are the two terms inside the binomial.
The exponent $ n $, which is the power to which the binomial is raised.
And finally, the terms $ T_k = \binom{n}{k} a^{n-k} b^k $ involve binomial coefficients and powers of $ a $ and $ b $.

This expansion is helpful because it simplifies the computation, eliminating the need to multiply the entire binomial by itself repeatedly. By identifying and substituting values for $ a $, $ b $, and $ n $, and then calculating the terms separately, you can expand complex binomials into manageable polynomial expressions.

Binomial Coefficient

Binomial coefficients play a crucial role in expanding expressions using the Binomial Theorem. They are denoted as $ \binom{n}{k} $ and can be calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, $ n! $ denotes the factorial of $ n $, which is the product of all positive integers up to $ n $. The binomial coefficient $ \binom{n}{k} $ gives the number of ways to choose $ k $ elements from a set of $ n $ elements without regard to order.

For each term in the binomial expansion, a specific binomial coefficient is used to calculate the correct term.
The term $ \binom{n}{k} $ represents how the exponents of $ a $ and $ b $ are distributed through the expansion process.
This coefficient ensures that each term is correctly balanced within the polynomial.

Understanding binomial coefficients allows for swift calculations in polynomial expansions, making it a valuable tool when working with the Binomial Theorem.

Polynomial Expression

A polynomial expression is a mathematical expression made up of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. These expressions can vary from a simple combination of terms to more complex formulas.

After applying the Binomial Theorem, the expanded form of a binomial becomes a polynomial expression.
The expression takes the form of multiple terms added together, each consisting of constant coefficients, variables, and powers.
In the expansion process, each term's degree is based on the sum of the exponents of its variables, such as $ x $ and $ y $, in our example.

For instance, in the exercise's solution, the expanded polynomial consists of terms like $ \frac{1}{x^5} $ and $ 243y^5 $, showcasing how fundamental parts of the binomial are transformed into individual terms of a complex polynomial. Understanding polynomial expressions is essential in algebra for simplifying, solving, and graphing equations.

91影视

For the following exercises, use the Binomial Theorem to expand each binomial. $$\left(\frac{1}{x}+3 y\right)^{5}$$

Short Answer

Step by step solution

Understand the Binomial Theorem

Identify Parameters

Calculate Each Term Using the Binomial Theorem

Write the Expanded Form

Key Concepts

Binomial Expansion

Binomial Coefficient

Polynomial Expression

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