Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Use the binomial theorem to expand and simplify. $$ (x+y)^{6} $$

Step-by-Step Solution

Answer

The expansion is \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Step 1: Identify Parameters

In the binomial theorem, the expression \((x+y)^n\) is expanded using the formula: \( \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \). In this case, \(x+y\) is our binomial, and \(n = 6\).

Step 2: Use the Binomial Coefficient Formula

The binomial coefficient \(\binom{n}{k}\) is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). For each term, we'll compute the coefficient based on the current \(k\) value. For \(n=6\), this will be done for \(k\) ranging from 0 to 6.

Step 3: Expand Each Term

Compute each term with the following logic: the \(k^{th}\) term will be \(\binom{6}{k} x^{6-k} y^{k}\). For instance, for \(k=0\), the term is \(\binom{6}{0} x^{6} y^{0} = x^6\). Continue this process for all values of \(k\) up to 6.

Step 4: Calculate the Sum of Terms

Put together each calculated term from Step 3 to write the complete expansion of the binomial. So we have: \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Step 5: Verify and Simplify (if Needed)

Ensure all terms correctly use the binomial coefficient and exponents add up to 6. The expansion is already simplified in this standard form.

Key Concepts

Binomial Coefficient

In mathematics, particularly in combinatorics, understanding the binomial coefficient is essential. It is a key concept when dealing with polynomial expansions such as \((x+y)^6\), using the binomial theorem. The binomial coefficient, represented as \(\binom{n}{k}\), indicates the number of ways to choose \(k\) elements from a set of \(n\) elements. It is calculated by this formula:

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

Here, \(n!\) (n factorial) means multiplying all whole numbers from \(n\) down to 1. By using this formula, we can determine the coefficients of each term in the expansion. For our case:

• With \(n = 6\), the coefficients were calculated for \(k = 0\) to \(6\).
• These include terms such as \(\binom{6}{0}, \binom{6}{1}\), all the way to \(\binom{6}{6}\).

Each calculated coefficient plays a crucial role in multiplying with the respective powers of \(x\) and \(y\). As a result, it brings each term of the expansion to life.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a power of a binomial such as \((x+y)^n\) as a sum of terms. This is done using the binomial theorem, which provides a straightforward way to expand expressions like our original exercise \((x+y)^6\).

• The expansion involves computing terms \(x^{n-k}y^{k}\).
• Here, \(k\) is a variable integer that runs from 0 to \(n\).

In practice, you handle each term one by one. For \((x+y)^6\):

• Start with \(x^6\), then calculate succeeding terms like \(6x^5y\), \(15x^4y^2\), and so forth.
• This method ensures all possible products of \(x\) and \(y\), with their exponents adding up to 6, are included.

With polynomial expansion, intricate expressions can be represented in simpler terms, making complex algebraic manipulation more manageable.

Exponents

Exponents play a significant role when expanding expressions in mathematics, especially in polynomial expansions using the binomial theorem. They indicate repeated multiplication of a base number, enabling succinct expression of large calculations.

• An expression like \((x+y)^6\) uses the power 6 as an exponent.
• In the expansion process, \(\sum_{k=0}^{n} \binom{6}{k} x^{6-k} y^k\), the exponents and their roles become apparent.

For each term:

• The term \(x^{6-k}y^{k}\) shows how each variable is multiplied by itself multiple times.
• The exponent \(6-k\) for \(x\) and \(k\) for \(y\) fluctuate throughout the expansion.

Grasping the concept of exponents allows us to structure and understand polynomial expansions systematically. Understanding their behavior in these calculations helps one manage and simplify expressions effectively.