Problem 20 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(x + 1\right)^6 \)

Step-by-Step Solution

Answer

\((x + 1)^6 = x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1\

Step 1: Understand the Binomial Theorem Form

The Binomial Theorem itself is represented by the formula: \[ (x + y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k} \] where \(\binom{n}{k}\) means 'n choose k', also known as the binomial coefficient.

Step 2: Setting up the Expression

In the expression \((x + 1)^6\), we can see that we have \(x\) and \(1\) acting like \(x\) and \(y\) from the formula, with \(n\) being \(6\). So, we can substitute these into the binomial theorem to start expanding the expression.

Step 3: Applying the Binomial Theorem

Upon substitution in the formula, we get: \[ (x + 1)^6 = \sum_{k=0}^{6}\binom{6}{k}x^{6-k}1^{k} \] This simplifies to: \[ = \binom{6}{0}x^6 + \binom{6}{1}x^5 + \binom{6}{2}x^4 + \binom{6}{3}x^3 + \binom{6}{4}x^2 + \binom{6}{5}x + \binom{6}{6} \]

Step 4: Calculating and Rearranging

After calculating the binomial coefficients, we get: \[ = 1x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 \] And this is the final, expanded form of the expression.

Key Concepts

Understanding Binomial Coefficients

The binomial coefficient is a fundamental component of the Binomial Theorem. You might have seen it symbolized as \( \binom{n}{k} \), which is read as "n choose k." This value represents the number of ways in which \( k \) items can be chosen from \( n \) items without regard to the order of selection.
Here is how it can be calculated:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

In this formula, \( n! \) ("n factorial") is the product of all positive integers up to \( n \), \( k! \) is the product up to \( k \), and \( (n-k)! \) is the product up to \( n-k \).
For example, when expanding \((x+1)^6\), the coefficients such as \( 1, 6, 15, 20, 15, 6, \) and \( 1 \) are calculated through this very principle.

Expansion of Binomials Using the Binomial Theorem

Expanding binomials using the Binomial Theorem is a structured way to break down expressions like \( (x+y)^n \). Instead of multiplying factors over and over, the Binomial Theorem provides a neat formula:
\[(x + y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}\]
Here's how we apply it:

• In \((x+1)^6\), \( x \) and \( 1 \) substitute into the positions of \( x \) and \( y \).
• Identify \( n \) as \( 6 \). Every term becomes a product of a binomial coefficient, a power of \( x \), and a power of \( y \).

This method quickly generates all terms in the expansion by systematically increasing the power of \( y \) and decreasing the power of \( x \). For the given exercise, apply these steps to get a detailed expansion.

Polynomial Expressions Explained

A polynomial expression is a mathematical expression that involves sums and products of variables and coefficients. These terms are composed of variables raised to whole-number exponents.
In the context of our expansion, once we have expanded the binomial \((x+1)^6\), each term produced (such as \( 1x^6, 6x^5, \ldots, 1 \)) is part of a polynomial.
Characteristics of polynomial expressions include:

• Being composed of terms denoted by \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \).
• Degrees of a polynomial are determined by the highest power of \( x \) present.

In our case, the highest degree is \( 6 \) as indicated by the term \( 1x^6 \). Polynomial expressions can have various applications depending on their form and the context.

The Role of Combinatorics in Binomial Expansions

Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of elements in sets. It's crucial in the calculation of binomial coefficients which make the Binomial Theorem possible.
When using combinatorics for binomial expansion:

• The coefficient \( \binom{n}{k} \) represents choosing \( k \) successes in \( n \) trials, relating to patterns in scenarios involving probability and decision trees.
• This is why the values \( 1, 6, 15, 20, 15, 6, \) and \( 1 \) arise naturally as combinatorial products.

Understanding these combinatorial principles gives deeper insight into why the terms in binomial expansions appear as they do, enriching the study of algebra with real-world applications.