Problem 17 - Algebra 2

Question

Use the Binomial Theorem to expand each binomial. $$ (x-y)^{4} $$

Step-by-Step Solution

Answer

The expanded form of \((x-y)^4\) is \(x^4-4x^3y+6x^2y^2-4xy^3+y^4\).

Step 1: Identify the Coefficients and Terms

From the binomial theorem, the coefficients for the 4th row of Pascal's Triangle are 1, 4, 6, 4, 1. The binomial is \((x-y)^4\), so the first term is x and the second term is -y.

Step 2: Expand the Binomial

Start with the highest power of the first term (x) and no power of the second term (-y). As you move along, lower the power of the first term and raise the power of the second term. For \(n=4\), there are \(n+1=5\) terms: \[_{n=0}C_4*x^{4-0}*(-y)^0, _{n=1}C_4*x^{4-1}*(-y)^1, _{n=2}C_4*x^{4-2}*(-y)^2, _{n=3}C_4*x^{4-3}*(-y)^3, _{n=4}C_4*x^{4-4}*(-y)^4.\]

Step 3: Simplify each Term

Simplify each term, remembering to alternate the sign according to the power of -y. This results in: \[1x^4-4x^3y+6x^2y^2-4xy^3+1y^4.\]

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating and useful tool in mathematics, especially in combinatorics and binomial expansions. It is a triangular array of numbers where each number is the sum of the two numbers directly above it from the previous row. The rows are indexed starting from 0, and each row represents the coefficients in the binomial expansion of (a+b)^n .
For instance, the 4th row of Pascal's Triangle is 1, 4, 6, 4, 1. This row provides the coefficients for expanding (x-y)^4.
Here is a quick look at how it works:

• Row 0: 1
• Row 1: 1, 1
• Row 2: 1, 2, 1
• Row 3: 1, 3, 3, 1
• Row 4: 1, 4, 6, 4, 1

Each coefficient in the expansion can be matched with a term of the form (x-y)^4 , by multiplying these coefficients with the appropriate powers of (x) and (-y) , making the expansion process straightforward and systematic.

Binomial Expansion

Binomial Expansion refers to the process of expanding expressions that are raised to a power, such as (a+b)^n. The Binomial Theorem is a powerful formula that helps to expand these expressions without directly multiplying them out fully.
According to the Binomial Theorem, the expansion of a binomial expression like (x-y)^n is given by:
\((x-y)^n =\sum_{k=0}^{n}\binom{n}{k} x^{n-k} (-y)^k \)
This formula tells us that each term in the expansion consists of:

• A binomial coefficient \(\binom{n}{k}\), which can be found using Pascal's Triangle.
• The first term (x) raised to the power of (n-k).
• The second term (-y) raised to the power of (k)

Notice the pattern of increasing powers of (-y) and decreasing powers of (x), which explains why the expansion becomes longer as (n) increases. Understanding these patterns can help you tackle even larger binomial expressions efficiently.

Polynomial Expansion

Polynomial expansion focuses on expressing a polynomial raised to a power in terms of its individual components. When dealing with polynomials like (x-y)^4, the expansion process involves breaking it down into the sum of terms, each with its own distinct factors and coefficients.
To properly perform a polynomial expansion using the Binomial Theorem, here’s the general approach:

• Identify the polynomial or binomial expression you want to expand.
• Use the appropriate coefficients from Pascal's Triangle to construct each term.
• Combine powers of the variable terms (x)and (-y) according to the binomial theorem.
• Simplify the resulting expression by performing any necessary arithmetic.

In the example (x-y)^4, the expansion results in \(x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4\). Each term in this final result can be seen as a product determined using specific powers of (x) and (-y), carefully balanced by the coefficients from Pascal's Triangle to produce a complete polynomial form.