Problem 1 - Algebra and Trigonometry

Question

\(1-12\) . Use Pascal's triangle to expand the expression. $$ (x+y)^{6} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the row in Pascal's triangle

To find the coefficients of the binomial expansion of \((x+y)^6\), first identify the 7th row of Pascal's Triangle (since we start counting from row 0). This row is: \(1, 6, 15, 20, 15, 6, 1\).

Step 2: Apply the binomial theorem

The binomial theorem states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\). Using the coefficients from Step 1, the expansion is \(1 \cdot x^6 + 6 \cdot x^5 y + 15 \cdot x^4 y^2 + 20 \cdot x^3 y^3 + 15 \cdot x^2 y^4 + 6 \cdot x y^5 + 1 \cdot y^6\).

Step 3: Write the expanded form

Combine the terms with their coefficients to write out the full expansion: \(x^6 + 6x^5 y + 15x^4 y^2 + 20x^3 y^3 + 15x^2 y^4 + 6x y^5 + y^6\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers with wide applications in combinatorics, algebra, and probability theory. Each number in the triangle is the sum of the two numbers directly above it. It is used to find coefficients in the expansion of a binomial expression like \(x + y\)^n.

To locate the coefficients for a specific expansion, look at the row number that corresponds to the binomial's exponent. For example, if you're expanding \(x+y\)^6, you use the 7th row (remembering to start counting from 0) of Pascal’s Triangle. This row contains the coefficients [1, 6, 15, 20, 15, 6, 1].

• Start from the top at row 0 with the number 1.
• Each subsequent row begins and ends with 1.
• Inside numbers are derived from adding the two numbers above.

These coefficients are crucial for constructing the polynomial terms in a binomial expansion.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions that are raised to a power. It states that the expansion of \(x+y\)^n is:\[\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]where \( \binom{n}{k} \) are binomial coefficients.

These coefficients can be obtained from Pascal's Triangle or calculated using factorials: \ \binom{n}{k} = \frac{n!}{k!(n-k)!} \.

• The index \(k\) runs from 0 to \(n\).
• Each term is composed of a coefficient, a power of \(x\), and a power of \(y\).
• For \(x+y\)^6, the terms will range from \(x^6\) to \(y^6\) with decreasing and increasing powers respectively.

The Binomial Theorem helps quickly expand polynomials without manually multiplying out the entire expression.

Polynomial Expansion

Polynomial expansion involves expressing a binomial like \(x+y\)^n as a sum of terms. Each term in the expansion is a result of one of the combinations of the base terms raised to powers that sum to n.

In \(x+y\)^n, the coefficients of the expansion are determined using either the Binomial Theorem or directly from Pascal’s Triangle. Once you have the coefficients, you distribute them across each combination of \(x\) and \(y\) raised to the appropriate powers.

• Begin with the highest power of \(x\) and reduce it step by step.
• Simultaneously increase the power of \(y\) to maintain the total power as \(n\).
• Make sure all terms include the related coefficient from Pascal's Triangle.

Separate the terms with addition and ensure all calculations maintain the original equation’s properties. This method organizes the expansion process and avoids common mistakes in manual multiplication of binomials.