Problem 27 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x-y)^{5} $$

Step-by-Step Solution

Answer

The simplified form of the binomial \( (3x - y)^5 \) is \( 243x^5 - 405x^4y + 270x^3y^2 - 90x^2y^3 + 15xy^4 - y^5 \).

Step 1: Recall the Binomial Theorem

The Binomial Theorem is represented by the formula: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k}b^k \), where \(\binom{n}{k}\) is a binomial coefficient.

Step 2: Identify the variables

In the given binomial \( (3x - y)^5 \), the variables are as follows: \( a = 3x \), \( b = -y \), and \( n = 5 \).

Step 3: Apply the Binomial Theorem and Expand

Substitute the values into the binomial theorem formula: \[ (3x - y)^5 = \binom{5}{0}(3x)^5(-y)^0 - \binom{5}{1}(3x)^4(-y)^1 + \binom{5}{2}(3x)^3(-y)^2 - \binom{5}{3}(3x)^2(-y)^3 + \binom{5}{4}(3x)^1(-y)^4 - \binom{5}{5}(3x)^0(-y)^5 \]

Step 4: Simplify the Result

Simplify each term: \[ 243x^5 - 5 * 81x^4y + 10 * 27x^3y^2 - 10 * 9x^2y^3 + 5 * 3xy^4 - y^5 \]

Key Concepts

Understanding Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power. The expression \((a + b)^n\) is expanded into several terms that involve coefficients, powers of \(a\), and powers of \(b\). When you have a binomial like \((3x - y)^5\), the process of binomial expansion helps rewrite this expression as a sum of simpler terms.

• Each term in the expansion is derived from multiplying appropriate powers of the two terms \(a\) and \(b\) in the binomial.
• The binomial expansion involves terms that decrease in powers of \(a\) and increase in powers of \(b\).
• For each term, you'll multiply a binomial coefficient with the appropriate powers of \(a\) and \(b\).

Understanding how to apply this method will allow you to transform complex algebraic expressions into manageable pieces.

Decoding the Binomial Coefficient

The binomial coefficient is an essential part of the binomial expansion. It's represented by \(\binom{n}{k}\), which is read as "n choose k". This coefficient determines how many ways you can choose \(k\) elements from \(n\) elements. These coefficients give weight to each term in the binomial expansion and can be calculated using combinations.

• The binomial coefficient \(\binom{n}{k}\) is calculated as \(\frac{n!}{k!(n-k)!}\), where \(!\) denotes the factorial function.
• For example, in the expansion of \((3x-y)^5\), if \(n=5\) and \(k=2\), then \(\binom{5}{2} = 10\).
• These coefficients appear naturally in Pascal's Triangle, which is a quick and visual way to determine them.

Binomial coefficients help distribute the effects across each term consistently as the binomial is expanded.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the goal after performing a binomial expansion. This involves combining like terms and reducing any arithmetic to achieve a more simplified result. For the example \((3x - y)^5\):

• Each term of the expansion like \(243x^5\), \(-405x^4y\), to \(-y^5\) is analyzed.
• You perform calculations such as \(5 \times 81 = 405\) to get coefficients for terms.
• Once calculations are done, you look for similar terms that can be combined for further simplification, ensuring that variables are expressed in their simplest forms.

Through this simplification process, complex expressions become more manageable and are easier to work with.