Problem 5 - Precalculus Mathematics for Calculus

Question

Use Pascal's triangle to expand the expression. $$(x+y)^{6}$$I

Step-by-Step Solution

Answer

Expand to: \(x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Step 1: Understand Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The rows in Pascal's Triangle correspond to the coefficients in the binomial expansion. For the expansion of \((x+y)^n\), we will need the coefficients from the \((n+1)\)-th row of Pascal's Triangle. Here, \(n=6\), so we will use the 7th row.

Step 2: Identify the 7th Row of Pascal's Triangle

The 7th row of Pascal's Triangle is: \(1, 6, 15, 20, 15, 6, 1\). These numbers are the coefficients for the expansion of \((x+y)^6\).

Step 3: Write the General Binomial Expansion Formula

The formula for expanding \((x+y)^n\) is: \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). This means for each term, you take \(x\) raised to the power \((n-k)\) and \(y\) raised to the power \(k\), multiplied by the binomial coefficient \(\binom{n}{k}\).

Step 4: Apply the Coefficients to the Expansion

Using the coefficients \(1, 6, 15, 20, 15, 6, 1\), substitute into the binomial expansion: \((x+y)^6 = 1x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + 1y^6\).

Step 5: Simplify the Expression

Combine and simplify each term if necessary: \((x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6\).

Key Concepts

Binomial Expansion

The binomial expansion is a way to expand expressions that are raised to a power, using binomials which are polynomials with two terms like (x + y)^n. This method is essential for simplifying expressions and finding powers of sums. Instead of multiplying out (x + y) repeatedly, the binomial expansion allows us to expand it systematically using coefficients from Pascal's Triangle.
For example, when given ((x+y)^6) , rather than multiplying (x+y) five more times, you find the expansion as : (x+y)^6 = 1x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6.

• The terms involve powers of x decreasing from 6 to 0, and powers of y increasing from 0 to 6.
• Binomial terms are organized such that each term is a product of a binomial coefficient, a power of x , and a power of y.

This pattern makes computation much easier, especially for larger powers.

Coefficients

Coefficients in the binomial expansion are numbers that are calculated using Pascal's Triangle or the binomial coefficient formula. In our exercise, the coefficients come from the 7th row of Pascal's Triangle which are: 1, 6, 15, 20, 15, 6, 1.

• The coefficients indicate how many ways you can choose elements from a set, also known as combinations.
• For the expression (x+y)^6, each coefficient multiplies the terms x^n-k y^k.

This means that the coefficient gives each term in the expansion its weight, making the expansion more manageable and accurate.
With Pascal's Triangle or by using \( \binom{n}{k} \), you can find these crucial coefficients without having to calculate each possible combination individually.

Binomial Theorem

The Binomial Theorem provides a formulaic way to expand binomials raised to a power. It states:\[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
The theorem helps us understand the structure of binomial expansions and see patterns among the coefficients. Using this theorem, you can pinpoint exactly how to expand any binomial raised to a certain power without needing to manually calculate each term.

• Every term in the expansion involves a binomial coefficient which is derived from either the Binomial Theorem or Pascal’s Triangle.
• The terms exhibit symmetry: the k-th term's xexponent, (n-k), diminishes while y'sexponent, k, increases.

By grasping the Binomial Theorem, you can efficiently tackle complex algebraic expansions, making it a powerful tool in both basic and advanced mathematics.