Problem 50 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{2}+x^{-3}\right)^{4} $$

Step-by-Step Solution

Answer

The expansion and simplification of the given binomial expression is \(x^{8} + 4x^{5} + 6x^{2} + 4x^{-1} + x^{-4}\). This answer has been obtained by applying the binomial theorem and simplifying the terms.

Step 1: Identify the Elements in the Binomial Theorem

In the expression \((x^{2}+x^{-3})^{4}\), \(x^{2}\) corresponds to 'x', \(x^{-3}\) corresponds to 'y', and the exponent '4' corresponds to 'n' in the binomial theorem.

Step 2: Use the Binomial Theorem to Expand

Using the Binomial theorem and summing from k=0 to 4, the terms will be \({\binom{4}{0}} (x^{2})^{4-0} (x^{-3})^{0} + {\binom{4}{1}} (x^{2})^{4-1} (x^{-3})^{1} + {\binom{4}{2}} (x^{2})^{4-2} (x^{-3})^{2} + {\binom{4}{3}} (x^{2})^{4-3} (x^{-3})^{3} + {\binom{4}{4}} (x^{2})^{4-4} (x^{-3})^{4}\)

Step 3: Calculate the Binomial Coefficients and Simplify

The coefficients \({\binom{4}{0}}, {\binom{4}{1}}, {\binom{4}{2}}, {\binom{4}{3}}, {\binom{4}{4}} = 1, 4, 6, 4, 1\) respectively, and the expanded form simplifies to \(x^{8} + 4x^{5} + 6x^{2} + 4x^{-1} + x^{-4}\).

Key Concepts

Binomial Coefficients

Binomial coefficients are integral to understanding the binomial theorem, which is a formula describing the algebraic expansion of powers of a binomial expression. A binomial expression is an algebraic expression that contains two terms, such as \(a + b\).

The binomial coefficients appear in the binomial expansion as the numerical factors that multiply the variable terms. They can be found in Pascal's Triangle or calculated using the formula \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\], where \(n!\) denotes the factorial of \(n\), \(k\) is the term index, and \(k!\) and \(n-k!\) are the factorials of \(k\) and \(n-k\), respectively.

In the expansion of \(\left(x^{2}+x^{-3}\right)^{4}\), the coefficients answer the question of how many ways there are to choose a certain number of \(x^{2}\) terms out of the four available when expanding the binomial expression. These coefficients, calculated as \(1, 4, 6, 4, 1\) for the corresponding powers, reflect the symmetry of Pascal's Triangle.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. Letters are used to represent variables or unknown values that can take on different numbers.

An important property of algebraic expressions is that they can be simplified or manipulated using algebraic rules and operations. For instance, \(\left(x^{2}+x^{-3}\right)^{4}\) is a complex algebraic expression, which can be expanded using the binomial theorem to form a series of simpler terms that are more easily evaluated or simplified further.

Importance of Algebraic Expression in Binomial Expansion

Understanding how to form and work with algebraic expressions is vital when dealing with the binomial theorem. Each term in the expansion depends on correctly applying powers to the variables in the expression and determining the new coefficients that multiply these variable terms.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its most basic form without changing its value. This often involves combining like terms, factoring, expanding expressions, and canceling terms when possible. Simplifying can make expressions easier to understand and work with, especially when solving equations or evaluating expressions.

In the context of the binomial theorem, after expansion, the expression will often contain terms that can be simplified. Taking the exercise \(\left(x^{2}+x^{-3}\right)^{4}\), after applying the binomial theorem and expanding, we obtained an expression with various powers of \(x\). To simplify it, we combined coefficients with the appropriate powers of \(x\), applied the rules of exponents, and finally arranged the terms in descending order of their exponents to achieve the simplified expression \(x^{8} + 4x^{5} + 6x^{2} + 4x^{-1} + x^{-4}\).

Role in Clear Communication

Simplifying expressions not only helps in solving problems but also in the clear communication of mathematical ideas. A simplified expression is more accessible and easier to interpret, both in written and verbal communication.