Problem 102 - Algebra 2

Question

Expand each binomial. $$ (x+y)^{6} $$

Step-by-Step Solution

Answer

\(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \)

Step 1: Identify the binomial

The binomial given in the problem is \( (x+y) \) and it is raised to the power 6.

Step 2: Recognize the pattern of coefficients using Pascal's triangle

Using Pascal's triangle or the binomial theorem formula \( ^nC_k \), we find that the coefficients for an expression raised to the power 6 are 1, 6, 15, 20, 15, 6, 1.

Step 3: Apply the power rule

Now, we apply the power rule to each component, decreasing the power of x and increasing the power of y. x starts with power 6 and y with power 0, and for each subsequent term, we decrease the power of x by 1 and increase that of y by 1.

Step 4: Expand the binomial

Now, combine the coefficients, the variables \(x\) and \(y\), and their correct powers to expand the binomial: \( (x+y)^6 = 1x^6y^0 + 6x^5y^1 + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6x^1y^5 + 1x^0y^6 \)

Step 5: Simplify the expanded expression

The term \(x^0\) is 1 and so is \(y^0\). Therefore, the expanded binomial expression becomes \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \)

Key Concepts

Pascal's Triangle

Pascal's Triangle is an easy way to find coefficients for binomial expansions. Start with a row of 1 at the top, then build by adding numbers above and to the left. If you need the 6th row for example (used in our problem), begin with:

• 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
• Row 5: 1, 5, 10, 10, 5, 1
• Row 6: 1, 6, 15, 20, 15, 6, 1

In our example, these numbers—1, 6, 15, 20, 15, 6, 1—are the coefficients for \((x + y)^6\). This pattern grows infinitely, making it incredibly powerful for expanding any binomial.

Binomial Theorem

The Binomial Theorem is a formula that provides a shorthand way of expanding binomials. Using it, we can find each term of a binomial raised to a power without having to multiply the entire expression out.
For a binomial \((a + b)^n\), the theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\]This formula uses:

• \(\binom{n}{k}\), which is a binomial coefficient found using Pascal's Triangle or calculated with factorials
• The variable \(a\) decreasing in exponent from \(n\) to 0
• The variable \(b\) increasing in exponent from 0 to \(n\)

In the expansion of \((x + y)^6\), each term uses these coefficients with powers of \(x\) dropping from 6 to 0 and powers of \(y\) rising from 0 to 6.

Polynomial Expansion

Polynomial Expansion is the process of multiplying out expressions raised to a power, making each term visible as in our problem. For the binomial \((x+y)^6\), we use polynomial expansion to rearrange and combine terms.
Each term in the expansion \((x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\) is formed by multiplying coefficients from Pascal's Triangle with the powers of \(x\) and \(y\), as found using the Binomial Theorem.

• Start from the left: highest power of \(x\) and lowest of \(y\)
• Move right: decrease \(x\)'s power by 1, increase \(y\)'s power by 1
• Repeat until you reach the end

Each combination gives one term of the polynomial, resulting in a fully expanded form.