Problem 48 - College Algebra

Question

In Exercises 39-48, find the term indicated in each expansion. $$(x+2 y)^{10} ;$$ the term containing \(y^{6}\)

Step-by-Step Solution

Answer

The term containing \(y^{6}\) in the expansion of \((x+2y)^{10}\) is \(210 * x^4 * 64y^{6}\)

Step 1: Identify the correct k

We need to find the term with \(y^{6}\). In the binomial theorem, \(b^k\), the power of \(b\), corresponds to the \(k\)'th term. So, for our problem, \(k=6\) because \(y\) takes the place of \(b\), and we want the term containing \(y^{6}\).

Step 2: Apply the Binomial theorem

We substitute \(k=6\), \(a=x\), \(b=2y\), and \(n=10\) into the binomial theorem. This gives us the 6th term \( ^{10}C_{6} * x^{10-6} * (2y)^6 \).

Step 3: Evaluate the term

Evaluating \( ^{10}C_{6} \) gives 210, and \(x^{10-6}\) simplifies to \(x^4\). Then, \((2y)^6\) is equal to 64\(y^{6}\). So, the term is \(210 * x^4 * 64y^{6}\).

Key Concepts

Binomial Expansion

The Binomial Theorem is a powerful algebraic tool used to expand expressions that are raised to a power, such as \((x + y)^n\). This theorem reveals how every binomial, through expansion, results in multiple terms. These terms comprise the elements of the original binomial raised to various powers.
One of the best aspects of the binomial theorem is that it simplifies the computation of specific terms within these raised expressions. It provides a formula for the expansion:\[(a + b)^n = \sum_{k=0}^{n} { {n \choose k} a^{n-k} b^k }\]
Each component within this formula has its purpose:- \({n \choose k}\) is the binomial coefficient, which tells us how many ways we can select elements.- \(a^{n-k}\) is derived from the first component of the binomial, diminishing as power increases.- \(b^k\) results from the second component, elevating as power increases.
In essence, binomial expansion utilizes the above formula to keep calculations straightforward, especially when focusing on finding specific terms within the binomial's expansion.

Combinatorics

Combinatorics is a fascinating branch of mathematics dealing with the counting, arrangement, and combination of objects. It plays a crucial role in the binomial theorem through the use of binomial coefficients. These coefficients, symbolized as \({n \choose k}\), express how many ways \(k\) elements can be chosen from a total of \(n\) elements without considering the order.
Let's break down the formulas:- The term \({n \choose k}\) is formally written as \[\frac{n!}{k!(n-k)!}\]- Here, \(n!\) is the factorial of \(n\), meaning the product of all integers from 1 to \(n\).
Binomial coefficients are important because:

• They determine the weight of each term in the binomial expansion.
• They help in identifying how many ways selections can be made from a larger set.

In practice, knowing how to compute these coefficients assists in solving challenging problems not just in binomial expansions but also in permutations and combinations.

Algebraic Expressions

Algebraic expressions are fundamental components in mathematics. They are composed of variables, numbers, and operations such as addition, subtraction, multiplication, and division.
Understanding algebraic expressions is key to grasping concepts like the binomial theorem because:

• They form the basis for constructing and simplifying mathematical formulas.
• They allow for manipulation and transformation to find unknown values.

When dealing with an expression like \((x + 2y)^{10}\), you interpret the expression via its components:x and y are variables, while 2 is a constant. Raising the entire expression to the 10th power requires dealing with intricate computations made simple using techniques like the binomial theorem.
Harnessing the power of algebraic expressions allows us to establish and solve equations which can model real-world phenomena. They exist at the heart of much mathematical analysis from simple arithmetic to complex number theory.