Question

Using the Binomial Theorem, expand: (a) \((x+y)^{3}\) (b) \((x+y)^{6}\)

Short answer

(a) \((x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\); (b) \((x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem 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. This allows us to expand any binomial expression \((x+y)^n\) as a series of terms.

Expansion of \((x+y)^3\)

Using the Binomial Theorem, expand \((x+y)^3\): - \((x+y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} y^k\)Substituting the values of \(k\) from 0 to 3:- For \(k=0\), the term is \(\binom{3}{0}x^{3-0}y^{0} = 1 \cdot x^3 = x^3\)- For \(k=1\), the term is \(\binom{3}{1}x^{3-1}y^{1} = 3 \cdot x^2y\)- For \(k=2\), the term is \(\binom{3}{2}x^{3-2}y^2 = 3 \cdot xy^2\)- For \(k=3\), the term is \(\binom{3}{3}x^{3-3}y^3 = 1 \cdot y^3 = y^3\)Thus, \((x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\).

Expansion of \((x+y)^6\)

Using the Binomial Theorem, expand \((x+y)^6\): - \((x+y)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} y^k\)Substituting the values of \(k\) from 0 to 6:- For \(k=0\), the term is \(\binom{6}{0}x^{6-0}y^{0} = 1 \cdot x^6 = x^6\)- For \(k=1\), the term is \(\binom{6}{1}x^{6-1}y^{1} = 6 \cdot x^5y\)- For \(k=2\), the term is \(\binom{6}{2}x^{6-2}y^2 = 15 \cdot x^4y^2\)- For \(k=3\), the term is \(\binom{6}{3}x^{6-3}y^3 = 20 \cdot x^3y^3\)- For \(k=4\), the term is \(\binom{6}{4}x^{6-4}y^4 = 15 \cdot x^2y^4\)- For \(k=5\), the term is \(\binom{6}{5}x^{6-5}y^5 = 6 \cdot xy^5\)- For \(k=6\), the term is \(\binom{6}{6}x^{6-6}y^6 = 1 \cdot y^6 = y^6\)Thus, \((x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Review and Finalize

Check our calculations to ensure each term is accurate by verifying the binomial coefficients and matching each term with its expression format \(\binom{n}{k} x^{n-k} y^k\). Both expansions as derived are correct.

Key concepts

Binomial Coefficients

Binomial coefficients play a crucial role in the binomial theorem, which is a fundamental concept in algebra. These coefficients are the numbers found in the expansion of a binomial raised to a power. They are expressed as \(\binom{n}{k}\), which reads as "n choose k." For any non-negative integer \(n\), and integer \(k\) such that \(0 \leq k \leq n\), the binomial coefficient \(\binom{n}{k}\) is given by the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This formula measures how many different ways you can choose \(k\) elements from a set of \(n\) elements, which directly relates to counting and arranging combinations.

A few key points about binomial coefficients include:

• For every \(n\), \(\binom{n}{0} = 1\) and \(\binom{n}{n} = 1\).
• They are symmetric: \(\binom{n}{k} = \binom{n}{n-k}\).
• The sum of all binomial coefficients for a given \(n\) equals to \(2^n\) (which can be seen in the sum of the terms in the expansion of \((x+y)^n\)).

Understanding binomial coefficients helps simplify polynomial expansions, especially in applications like calculus, probability, and statistics.

Polynomial Expansion

Polynomial expansion using the binomial theorem involves expanding expressions like \((x + y)^n\) into a sum of terms. Each term in the expansion has the general form \(\binom{n}{k} x^{n-k} y^k\).

Let's break down the steps to expand a simple binomial:

• Decide the value of \(n\), which is the power to which the binomial is raised.
• For each k from 0 to n, calculate the binomial coefficient \(\binom{n}{k}\).
• Compute the term \(x^{n-k} y^k\) for each value of \(k\).
• Combine all these terms to form the full expanded polynomial.

Let's consider an example:
For \((x+y)^3\):- The expansion is given by summing over all values of \(k\), from 0 to 3:
\(x^3 + 3x^2y + 3xy^2 + y^3\).

The use of polynomial expansion is essential in many areas of mathematics, helping to simplify calculations and facilitate understanding of higher-degree polynomials.

Discrete Mathematics

Discrete mathematics is a vast field that includes the study of mathematical structures that are fundamentally discrete rather than continuous. This area encompasses a wide range of topics and techniques commonly used in computer science and mathematical proofs.

Binomial coefficients and polynomial expansion are significant subjects in discrete mathematics. They are used in:

• Combinatorics: For counting combinations and arrangements, such as calculating ways to choose items.
• Algorithm analysis: Particularly in understanding recursive algorithms and combinatorial structures.
• Probability theory: To determine the likelihood of various outcomes.
• Graph theory: Where counting paths between nodes may involve binomial coefficients.

Understanding these concepts deepens one's comprehension of discrete structures and helps in finding solutions for discrete problems. Such skills are invaluable in fields such as computer science, engineering, and operations research. The discrete nature of these topics makes them distinct from other branches of mathematics like calculus, which deals with continuous change.