Problem 28 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-3 y)^{5}$$

Step-by-Step Solution

Answer

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

Step 1: Apply the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case, \(a = x\), \(b = -3y\), and \(n = 5\). Applying the theorem, we get: \((x-3y)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} (-3y)^{k}\).

Step 2: Expand with Pascal's Triangle

Now expand each term in the equation by applying the coefficients from Pascal's triangle (specifically, the row starting with 1, 5). The expanded form is: \( \binom{5}{0}x^{5}(-3y)^0 + \binom{5}{1}x^{4}(-3y)^1 + \binom{5}{2}x^{3}(-3y)^2 + \binom{5}{3}x^{2}(-3y)^3 + \binom{5}{4}x(-3y)^4 + \binom{5}{5}(-3y)^5\).

Step 3: Simplify the Expression

Simplify each term, taking \(\binom{5}{k}\) as the coefficient, \(x\) raised to a power, and \((-3y)\) raised to a power as well. The simplified expression is \(x^5 - 15x^4y + 90x^3y^2 - 270x^2y^3 + 405xy^4 - 243y^5\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that provides coefficients for expanding binomials. Each number is the sum of the two directly above it, creating a beautiful pattern.

To construct it, start with a "1" at the top. The second row has "1"s on both ends, and each subsequent row begins and ends with "1"s too.

The numbers between the "1"s are determined by adding the two numbers above them. So, the third row is "1, 2, 1", the fourth is "1, 3, 3, 1", and so on.

For expanding a binomial expression like \((x-3y)^5\), we use the 5th row: "1, 5, 10, 10, 5, 1." These are the coefficients in our expansion.

• 1st term has coefficient 1
• 2nd term has coefficient 5
• 3rd term has coefficient 10
• 4th term also has coefficient 10
• 5th term has coefficient 5
• 6th term returns to 1

The triangle greatly simplifies finding coefficients for the binomial expansion without complex calculations.

Binomial Expansion

The binomial expansion uses the Binomial Theorem to expand expressions of the form \((a + b)^n\). This theorem helps us transform binomials into a sum of terms involving powers of \(a\) and \(b\).

Here's the formula: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). It uses the binomial coefficient \(\binom{n}{k}\), which is found using Pascal's Triangle or the combination formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

For \((x-3y)^5\), with \(n=5\), \(a=x\), and \(b=-3y\), the expansion becomes: \(\sum_{k=0}^{5} \binom{5}{k} x^{5-k} (-3y)^{k}\). Each term is constructed as a combination of powers of \(x\) and \(-3y\), multiplied by their respective coefficients from Pascal's Triangle or using the combination formula.

• 1st term: \(x^5 \times 1\)
• 2nd term: \(x^4 \times (-3y) \times 5\)
• and so on...

Expanding this way helps in creating an expanded form with terms that are easier to work with.

Algebraic Expressions

Algebraic expressions like \((x-3y)^5\) can be rewritten into multiple terms using expansion methods. These expressions are combinations of numbers, variables, and mathematical operations like addition or multiplication.

Simplification refers to the process of reducing the expression into a standard form. Once expanded using the Binomial Theorem or Pascal's Triangle, each term is simplified to make calculations or further algebraic manipulations easier.

In this context:

• Combine the coefficients from the Binomial Theorem with your variables, i.e. \(x\) and \(-3y\).
• Restructure powers as per their expansion result.
• Combine any like terms if necessary, to get the final expression.
  
  Thus, \((x-3y)^5\) simplifies to: \(x^5 - 15x^4y + 90x^3y^2 - 270x^2y^3 + 405xy^4 - 243y^5\). Each term clearly indicates a degree or power of \(x\) and \(y\) present in the original expression.

Understanding these simplifications can clarify how components interact in mathematical models and real-world applications.