Problem 27 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ \left(1-x^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form is \(1 - 4x^2 + 6x^4 - 4x^6 + x^8\).

Step 1: Identify the Binomial Expression

The given expression is \((1-x^2)^4\). This is a binomial expression where \(a = 1\) and \(b = -x^2\), with the power \(n = 4\).

Step 2: Write the Binomial Theorem Formula

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). We will use this formula to expand the given expression.

Step 3: Apply the Binomial Theorem

Substitute \(a = 1\), \(b = -x^2\), and \(n = 4\) into the formula: \[(1-x^2)^4 = \sum_{k=0}^{4} \binom{4}{k} (1)^{4-k} (-x^2)^k\].

Step 4: Calculate Each Term

Calculate each term \(\binom{4}{k} (1)^{4-k} (-x^2)^k\) for \(k = 0\) to \(4\):- For \(k = 0\): \(\binom{4}{0} (1)^4 (-x^2)^0 = 1\).- For \(k = 1\): \(\binom{4}{1} (1)^3 (-x^2)^1 = -4x^2\).- For \(k = 2\): \(\binom{4}{2} (1)^2 (-x^2)^2 = 6x^4\).- For \(k = 3\): \(\binom{4}{3} (1)^1 (-x^2)^3 = -4x^6\).- For \(k = 4\): \(\binom{4}{4} (1)^0 (-x^2)^4 = x^8\).

Step 5: Sum the Terms to Form the Expanded Expression

Add up all the terms from Step 4 to get the expanded expression: \(1 - 4x^2 + 6x^4 - 4x^6 + x^8\).

Key Concepts

Binomial Expansion

The idea of binomial expansion revolves around expanding an expression that is raised to a power, and it relies on the structure expressed by the Binomial Theorem. Generally, these expressions take the form \((a + b)^n\). In this concept, each term in the expansion is derived from the coefficients and powers of the terms in the binomial.For any binomial expression \((a + b)^n\), the Binomial Theorem provides a systematic method to expand it as:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This formula allows us to calculate each term of the expansion step by step. The symbols \(\binom{n}{k}\) are known as binomial coefficients, and they are key to determining the specific factors affecting each term.

• \(a\) is taken to the power of \((n-k)\)
• \(b\) is taken to the power of \(k\)
• Sum up all terms as \(k\) varies from 0 to \(n\)

This is especially useful for polynomial expressions, allowing us to break them down into simpler terms.

Pascal's Triangle

Pascal's Triangle is a powerful tool in mathematics that helps calculate binomial coefficients. It's essentially a triangular array of numbers, each number representing a binomial coefficient. To understand how it works, the top of the triangle starts with the number 1. Each number in the triangle is the sum of the two numbers directly above it. The triangle helps visualize and find the coefficients needed in the Binomial Theorem's formula.Here's a simple view of Pascal's Triangle's initial rows:- Row 0: \(1\)- Row 1: \(1, 1\)- Row 2: \(1, 2, 1\)- Row 3: \(1, 3, 3, 1\)- Row 4: \(1, 4, 6, 4, 1\) When expanding \((a + b)^n\), the coefficients for each term in the expansion correspond to the numbers in the \(n\)-th row of Pascal's Triangle. For example, for an expression like \((1-x^2)^4\), we use the 4th row: \(1, 4, 6, 4, 1\) as our binomial coefficients. This provides a quick reference and simplifies the calculation of the expansion. Pascal's Triangle not only simplifies the computation but also saves time when solving polynomial expansion problems.

Polynomial Expressions

Polynomial expressions are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They play an essential role in algebra and can include terms like \(x^2\), \(-3x\), or constants.The binomial expansion often results in a polynomial expression, where each resulting term is a part of the polynomial. For example, expanding \((1-x^2)^4\) gives us a polynomial expression \(1 - 4x^2 + 6x^4 - 4x^6 + x^8\).Breaking down polynomials into their simplest form requires combining like terms and arranging them in either ascending or descending order based on powers of the variables. The structure of a polynomial is similar to organizing large quantities of information into digestible parts, which aids in easier manipulation and understanding.

• Each term consists of coefficients and variables raised to various powers.
• Terms can be arranged in a sequence to represent the entire polynomial.
• Polynomials with higher degrees or powers can depict more complex relationships or solutions.

Understanding polynomials and how they relate to binomial expansions is crucial for solving more complex mathematical problems and can provide insights into systems modeled by these expressions.