Problem 11 - Algebra and Trigonometry

Question

Use Pascal’s triangle to expand the expression. $$ \left(x^{2} y-1\right)^{5} $$

Step-by-Step Solution

Answer

The expanded form is \(x^{10} y^5 - 5x^8 y^4 + 10x^6 y^3 - 10x^4 y^2 + 5x^2 y - 1\).

Step 1: Identify the Binomial Terms

The given expression is \((x^2 y - 1)^5\). We need to expand this expression using the binomial theorem, which states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Let \(a = x^2 y\) and \(b = -1\) with \(n = 5\).

Step 2: Write Pascal's Triangle for n=5

Pascal's triangle provides the binomial coefficients. For \(n = 5\), the row is: 1, 5, 10, 10, 5, 1. These will be the coefficients in the expanded form.

Step 3: Expand the Binomial Using Coefficients

Apply the binomial theorem: \((x^2 y)^5 (-1)^0\), \(5(x^2 y)^4 (-1)^1\), \(10(x^2 y)^3 (-1)^2\), \(10(x^2 y)^2 (-1)^3\), \(5(x^2 y)^1 (-1)^4\), and \((x^2 y)^0 (-1)^5\).

Step 4: Simplify Each Term

- \((x^2 y)^5 (-1)^0 = x^{10} y^5\)- \(5(x^2 y)^4 (-1)^1 = -5x^8 y^4\)- \(10(x^2 y)^3 (-1)^2 = 10x^6 y^3\)- \(10(x^2 y)^2 (-1)^3 = -10x^4 y^2\)- \(5(x^2 y)^1 (-1)^4 = 5x^2 y\)- \((x^2 y)^0 (-1)^5 = -1\)

Step 5: Combine the Terms

Combine all the simplified terms to get the expanded form: \(x^{10} y^5 - 5x^8 y^4 + 10x^6 y^3 - 10x^4 y^2 + 5x^2 y - 1\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that expands expressions raised to a power. It provides a formula to expand binomials, which are expressions involving the sum or difference of two terms. For any two numbers, \(a\) and \(b\), raised to a positive integer \(n\), the binomial theorem states:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This formula is a sum of terms involving binomial coefficients and powers of \(a\) and \(b\). The sum runs from \(k = 0\) to \(n\). In each term, the binomial coefficient \(\binom{n}{k}\) indicates how many ways to choose \(k\) elements from \(n\), and the powers \(a^{n-k}\) and \(b^k\) balance each other to always sum to \(n\).
The binomial theorem simplifies the otherwise lengthy process of multiplying binomials repeatedly. It is highly useful in combinatorics, probability, and various areas of mathematics and applied sciences.
In our original problem, the expression \((x^2 y - 1)^5\) is expanded using this theorem by assigning \(a = x^2 y\) and \(b = -1\). This allows us to systematically find each term of the expansion.

Binomial Expansion

Binomial expansion involves using the binomial theorem to express a binomial raised to a power as a series of terms. Each term in the expanded form results from applying the theorem's formula. Let's break it down:

• The expression \((a + b)^n\) gives us several terms, with each having specific coefficients and powers of \(a\) and \(b\).
• In our specific case, \((x^2 y - 1)^5\), the expansion involves terms like \(x^{10}y^5\), \(-5x^8y^4\), to \(-1\).

After substituting the values for \(a\) and \(b\), each term is calculated by:

• Determining the coefficient from Pascal's triangle or calculating \(\binom{n}{k}\).
• Raising \(a\) and \(b\) to their respective powers as determined by the binomial theorem.
• Multiplying these results together.

This process generates the full polynomial expression from the binomial, showcasing the efficiency of the binomial theorem for larger exponents.

Binomial Coefficients

Binomial coefficients are integral numbers that signify the number of ways to choose \(k\) elements from \(n\) elements without considering the order. They are denoted as \(\binom{n}{k}\) and play a central role in the binomial theorem.

• They appear as the crucial factors in the expansion of a binomial expression.
• In Pascal's triangle, these coefficients form the "entries" of each row, where each number is the combination of the two numbers directly above it.
• For example, the coefficients for \((x^2 y - 1)^5\) were 1, 5, 10, 10, 5, 1.

Calculating a binomial coefficient can be succinctly done using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes factorial, which is the product of all positive integers up to that number.
Pascal's triangle is a visual representation of these coefficients, and quickly provides them for moderate levels of \(n\). It allows us to shortcut the factorial calculations in problems like our example, where deriving the coefficients directly from the triangle is simpler and faster. This makes the binomial expansion process much more efficient.