Problem 8 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Expand \((x+y)^{5}\)

Step-by-Step Solution

Answer

The expansion is \(x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((x+y)^n\). It states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is a binomial coefficient.

Step 2: Identify Parameters

For the given expression \((x+y)^5\), we identify \(n = 5\). The expansion will have terms from \(k = 0\) to \(k = 5\).

Step 3: Calculate Binomial Coefficients

Use \(\binom{5}{k} = \frac{5!}{k!(5-k)!}\) to calculate the coefficients for each term from \(k=0\) to \(k=5\):\(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), \(\binom{5}{3} = 10\), \(\binom{5}{4} = 5\), \(\binom{5}{5} = 1\).

Step 4: Construct Each Term

Combine the coefficients with \(x\) and \(y\) to create each term:- For \(k=0\): \(\binom{5}{0} x^5 y^0 = x^5\)- For \(k=1\): \(\binom{5}{1} x^4 y^1 = 5x^4 y\)- For \(k=2\): \(\binom{5}{2} x^3 y^2 = 10x^3 y^2\)- For \(k=3\): \(\binom{5}{3} x^2 y^3 = 10x^2 y^3\)- For \(k=4\): \(\binom{5}{4} x^1 y^4 = 5x y^4\)- For \(k=5\): \(\binom{5}{5} x^0 y^5 = y^5\).

Step 5: Combine the Terms

Add all the terms calculated in the previous step to get the expanded form:\(x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\).

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are key components in the process of expanding binomials using the Binomial Theorem. These coefficients, often represented by \(\binom{n}{k}\), indicate the number of ways to pick \(k\) elements from a set of \(n\) elements. They are central in determining the weight of each term in the expanded expression.
When expanding \((x+y)^n\), each term in the expansion includes a binomial coefficient, calculated using the formula:

• \[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]

This formula breaks down as follows:

• \(n!\) (pronounced "n factorial") is the product of all positive integers from 1 to \(n\).
• \(k!\) is the product of all positive integers from 1 to \(k\).
• \((n-k)!\) is the product of all positive integers from 1 to \((n-k)\).

In the given example \((x+y)^5\), the binomial coefficients are calculated for each term from \(k = 0\) to \(k = 5\). This results in coefficients 1, 5, 10, 10, 5, and 1, respectively, each associated with a term in the expansion.

Breaking Down Mathematical Expansion

Mathematical expansion allows us to express a power of a binomial as a sum of terms involving the variables that make up the binomial. This is particularly useful when working with expressions like \((x+y)^n\), where a straightforward computation may be complex.
The Binomial Theorem provides a systematic method to expand such expressions by utilizing binomial coefficients. Each term in the expansion is a product of a binomial coefficient and the variables raised to relevant powers.

• The general form for each term is:
  \[\binom{n}{k} x^{n-k} y^k\]

The components of this term include:

• \(\binom{n}{k}\) – the binomial coefficient.
• \(x^{n-k}\) – the variable \(x\) raised to the power of \((n-k)\).
• \(y^k\) – the variable \(y\) raised to the power of \(k\).

This approach ensures that both variables in the binomial \((x+y)\) appear in each term, with the sum of their exponents always equaling \(n\). By following these steps for \(k\) from 0 to \(n\), we effectively construct the entire expanded expression.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that depict relationships and functions. In the realm of the Binomial Theorem, these expressions are enriched by a structure provided through expansion.
To understand the expression \((x+y)^n\), it helps to view it not merely as a repetition but as an arrangement governed by rules of algebra and mathematics. In general, algebraic expressions abide by simplification, substitution, and evaluation principles. For binomial expansion, a structured approach is introduced to untangle powers and coefficients neatly.

• In the case \((x+y)^5\), the expression unfolds into a sequence of terms:

\[x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\]
This expanded form showcases how algebra manipulates simple structures into more complex representations, capturing the interaction of variables and constants. By systematically determining each term in the context of its binomial coefficients and exponents, one can learn to handle even larger algebraic expressions with confidence. This process helps demystify algebra, making it more approachable and manageable for students.