Problem 56 - Algebra 2

Question

Find the specified term of each binomial expansion. Seventh term of \(\left(x^{2}-2 y\right)^{11}\)

Step-by-Step Solution

Answer

The seventh term of the \((x^{2}-2 y)^{11}\) binomial expansion is \(-29648x^{10}y^{6}\)

Step 1: Using the Binomial Theorem

According to the binomial theorem, any term 'k' in the binomial expansion of \((a+b)^n\) is given by: \[T_{k} = ^{n}C_{k-1} * a^{n-k+1} * b^{k-1}\]

Step 2: Substituting the values

In terms of the given expression \((x^{2}-2 y)^{11}\), we have \(a = x^{2}\), \(b = -2y\) and \(n = 11\). Moreover, we're looking for the seventh term, thus \(k = 7\). Substituting these values into the binomial theorem we get: \[T_{7} = ^{11}C_{7-1} * (x^{2})^{11-7+1} * (-2y)^{7-1}\]

Step 3: Simplifying the equation

After substituting the values, simplify the equation: \[T_{7} = ^{11}C_{6} * x^{10} * (-64y^{6})\] Using the combination theorem, \(^{11}C_{6}\) equals to 462. \[T_{7} = 462 * x^{10} * (-64y^{6}) = -29648x^{10}y^{6}\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool used to expand expressions of the form \((a+b)^n\), where \(a\) and \(b\) are any numbers or expressions, and \(n\) is a positive integer. This theorem not only helps find individual terms but also provides the entire expansion quickly.To understand the Binomial Theorem, we need a few basic notations:

• \(^{n}C_{k}\) represents "n choose k," a combinatorial way to calculate the number of ways to choose \(k\) items from \(n\) without replacement.
• Every term in the expansion is a product of powers of \(a\) and \(b\).
• The powers in each term of the expansion add up to \(n\).

The general term in the expansion is given by the formula: \[T_{k} = ^{n}C_{k-1} \cdot a^{n-k+1} \cdot b^{k-1}\] This formula allows us to efficiently compute any specific term without expanding an entire polynomial. It's particularly useful in problems where only a specific term is needed. This way, calculations are simplified significantly, saving time and effort.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing possible combinations or arrangements of elements. It plays a crucial role in the Binomial Theorem through the use of combinations, represented by \(^{n}C_{k}\).In our example, to find the specific term of the binomial expansion, we rely on combinatorial principles:

• The combination \(^{n}C_{k}\) calculates how many ways \(k\) elements can be chosen from a larger set of \(n\) elements.
• The formula for combinations is: \[^{n}C_{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes factorial, the product of an integer and all the integers below it down to 1.

Combinatorics is not only fundamental for binomial expansions, but it also appears in various fields like probability, optimizing strategies, and computer science algorithms. Mastery of this vital concept enhances mathematical understanding and problem-solving skills across different contexts.

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve sums of powers of variables combined with coefficients. For example, \((x^2 - 2y)\) is a polynomial expression where \(x^2\) and \(-2y\) are the terms.Key characteristics of polynomial expressions include:

• Terms: Each part of a polynomial that is added or subtracted. A term can be a constant, a variable, or a product of constants and variables raised to powers.
• Degree: The highest power of the variable present in the polynomial. The degree gives insight into the behavior and graph of the polynomial function.
• Coefficients: Numerical factors that multiply the variables or powers of variables within terms.

In the context of binomial expansions, a polynomial expression is expanded into multiple terms, each of which can be individually analyzed. Understanding polynomial expressions is essential for effectively utilizing the Binomial Theorem and simplifying expressions during the expansion process.

Algebraic Expressions

Algebraic expressions are foundational elements in algebra, consisting of numbers, variables, operations (like addition and multiplication), and possibly exponents. They express relationships and can represent anything from simple quantities to complex formulas. In binomial expansions, algebraic expressions become vital as:

• They are expanded to find specific terms or whole expressions using the Binomial Theorem.
• Simplifying and manipulating them requires an understanding of algebraic operations, such as factoring or distributing terms.
• Using algebraic expressions helps solve real-world problems through mathematical modeling.

Algebraic expressions allow abstract concepts to be portrayed in mathematical form, enabling evaluations, predictions, and the solving of equations. Develop a clear understanding of algebraic expressions to handle advanced topics and applications effectively.