Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \((x+1)^{6}\)

Step-by-Step Solution

Answer

The simplified and expanded form using the Binomial Theorem of \( (x+1)^6 \) is \( x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 \).

Step 1: Apply the Binomial Theorem

Now, apply the Binomial Theorem to the given expression, \( (x+1)^6 \). This implies that the expression will then be equivalent to \( Σ_{k=0}^6 (6 choose k) * x^(6-k) * 1^k \).

Step 2: Simplify the Resulting Series

Simplify the series by manually calculating for each term.

Step 3: Final Simplification and Result

Going through the series, we get this: the coefficient of \(x^6\) (for k = 0) is \( (6 choose 0) \) = 1, the coefficient of \(x^5\) (for k = 1) is \( (6 choose 1) \) = 6, the coefficient of \(x^4\) (for k = 2) is \( (6 choose 2) \) = 15, the coefficient of \(x^3\) (for k = 3) is \( (6 choose 3) \) = 20, the coefficient of \(x^2\) (for k = 4) is \( (6 choose 4) \) = 15, the coefficient of \(x\) (for k = 5) is \( (6 choose 5) \) = 6, the coefficient of \(x^0\) (for k = 6) is \( (6 choose 6) \) = 1.So, simplifying this gives, \( (x+1)^6 = x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 \).

Key Concepts

Binomial Coefficients

When dealing with binomial expressions raised to a power, the Binomial Coefficients play an essential role. They determine the coefficients of each term in the polynomial expansion.
A binomial coefficient is often expressed as \( \binom{n}{k} \), read as \( \text{"n choose k"} \), and it calculates the number of ways to choose \( k \) elements from a total of \( n \) elements without regard to the order of selection.
Mathematically, this is defined by the formula:

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

Here, \( n! \) (n factorial) represents the product of all positive integers up to \( n \), and the factorial operation \( k! \) and \( (n-k)! \) follow similarly.
In the binomial theorem, the coefficients will determine how each term in the expansion of a binomial like \((x +1)^6\) is formed. For instance, to expand \((x+1)^6\), each term involves these coefficients obtained via "6 choose k" as detailed in the original exercise solution.

Polynomial Expansion

The process of polynomial expansion involves expressing a binomial expression raised to a power as a sum of terms of different powers. In simpler terms, we take an expression like \((x+1)^6\) and convert it into a sequence of terms added together.
The Binomial Theorem facilitates this expansion. For a binomial expression \((x + a)^n\), it's represented as:

• \((x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k\)

Each term of this expansion is formed using a binomial coefficient, a power of \( x \) decreasing from \( n \) to \( 0 \), and a power of \( a \) that increases from \( 0 \) to \( n \).
For the specific example \((x+1)^6\), replacing \( a \) with 1, simplifies calculation since any power of 1 is just 1. The exercise demonstrates using these rules to expand and simplify this particular polynomial.

Combinatorics

Combinatorics is a fundamental area of mathematics largely concerned with counting, arrangements, and combinations. It provides the tools needed to understand and calculate binomial coefficients, which are central to the binomial theorem.
The problem of expanding \((x+1)^6\) benefits greatly from combinatorial insights. Understanding this, one sees how the counting methods used in combinatorics help illuminate why we use "6 choose k" when breaking down the binomial expression.
By applying combinatorial principles, one sees how to generalize and utilize these ideas across many different kinds of mathematical problems, particularly involving polynomial expansions and arrangements in sets. Combinatorics not only simplifies these calculations but also provides a clearer understanding of the underlying mathematical structures we work with in expansions.