Problem 13 - Precalculus

Question

In Exercises \(10-13,\) use Pascal's Triangle to expand the given binomial. $$ \left(x-x^{-1}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form is \(x^4 - 4x^2 + 6 - 4x^{-2} + x^{-4}\).

Step 1: Identify the Expansion Formula

The binomial theorem states that \((a+b)^n\) can be expanded using binomial coefficients: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\). For this problem, set \(a = x\), \(b = -x^{-1}\), and \(n = 4\).

Step 2: Find Binomial Coefficients Using Pascal's Triangle

Locate the row in Pascal's Triangle that corresponds to \(n=4\). The coefficients in this row are 1, 4, 6, 4, and 1.

Step 3: Write Each Term Using the Binomial Coefficients

The expansion of \((x-x^{-1})^4\) is given by: \(1 \cdot x^4 + 4 \cdot x^3(-x^{-1}) + 6 \cdot x^2(-x^{-1})^2 + 4 \cdot x(-x^{-1})^3 + 1 \cdot (-x^{-1})^4\).

Step 4: Simplify Each Term

Simplify each term of the expansion using algebraic manipulation:- \(1 \cdot x^4 = x^4\)- \(4x^3(-x^{-1}) = -4x^2\)- \(6x^2(x^{-2}) = 6\)- \(4x(x^{-3}) = -4x^{-2}\)- \(1 \cdot (x^{-4}) = x^{-4}\).

Step 5: Combine All Simplified Terms

Combine all the simplified terms to get the complete expanded form: \(x^4 - 4x^2 + 6 - 4x^{-2} + x^{-4}\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is an incredibly handy tool in mathematics. It is a triangular array of numbers, each line representing the coefficients of the binomial expansion. Each number is the sum of the two numbers directly above it. It begins with a '1' at the top, and each subsequent line adds two numbers together. Let's see how it helps in expanding binomials.

For example, for = 4, which is the fourth row, Pascal's Triangle gives us the sequence: 1, 4, 6, 4, and 1. These numbers are the coefficients when expanding a binomial raised to the fourth power.

• First number (1) represents the coefficient for the first term.
• The second number (4) is for the second term.
• Continuing, the third and fourth numbers (6 and 4) apply to the terms that follow.
• Finally, the last number (1) is the coefficient for the final term.

Pascal's Triangle simplifies finding coefficients, making it easier to write out and solve binomial expansions.

Binomial Coefficients

Binomial coefficients are a fundamental part of the binomial expansion. They are the numbers that appear in the rows of Pascal's Triangle and tell us how many different ways we can pick certain elements from a set. They are denoted as \(\binom{n}{k}\) and can be read as 'n choose k'.

When expanding a binomial such as \((a+b)^n\), the binomial coefficients define the weight that each specific combination contributes to the total expansion.

For instance, in the expansion of \((x-x^{-1})^4\), the coefficients from Pascal's triangle for \(n=4\) are 1, 4, 6, 4, and 1.

• For the first term, the coefficient 1 multiplies \(x^4\).
• The second term involves the coefficient 4, creating \(-4x^2\) after simplification.
• The middle term used 6 to simplify as 6, followed by another 4 for the next term, then ending with a 1 for the final term

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These coefficients determine the strength and placement of each term in the polynomial, ultimately affecting the overall expression.

Binomial Theorem

The Binomial Theorem is a powerful method for expanding binomials raised to any power. It provides a structured way of expressing \((a+b)^n\) as a sum, using binomial coefficients derived from Pascal's Triangle.

According to the theorem, \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\). This formula explains that each term of the expansion is a combination of powers of \(a\) and \(b\), multiplied by a corresponding binomial coefficient.

When applying the theorem to expand \((x-x^{-1})^4\), we see this process in action:

• Each term in the expansion comes from substituting into the formula.
• Each substitution calculates a term with its binomial coefficient, a power of \(a\), and a power of \(b\).
• Together, they form the expanded polynomial expression.

By using the Binomial Theorem, we can expand any binomial with ease, calculating each term efficiently and ensuring all coefficients match based on Pascal's guidelines.