Problem 26 - Precalculus Mathematics for Calculus

Question

Use the Binomial Theorem to expand the expression. $$(1-x)^{5}$$

Step-by-Step Solution

Answer

The expanded expression is \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem is used to expand expressions of the form \((a+b)^n\). It states that: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where \(\binom{n}{k}\) is the binomial coefficient and it's calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify Parameters

For the expression \((1-x)^5\), identify the parameters: \(a = 1\), \(b = -x\), and \(n = 5\).

Step 3: Expand Using the Binomial Theorem

Using the identified parameters, expand \((1-x)^5\): \[ (1-x)^5 = \sum_{k=0}^{5} \binom{5}{k} 1^{5-k} (-x)^k \]This simplifies to: \[ \sum_{k=0}^{5} \binom{5}{k} (-x)^k \]

Step 4: Calculate Each Term

Calculate each term using \( \binom{5}{k} \) for \(k = 0\) to \(5\):- For \(k = 0\): \(\binom{5}{0} (1)^5 (x)^0 = 1\)- For \(k = 1\): \(\binom{5}{1} (1)^4 (-x)^1 = -5x\)- For \(k = 2\): \(\binom{5}{2} (1)^3 (x)^2 = 10x^2\)- For \(k = 3\): \(\binom{5}{3} (1)^2 (-x)^3 = -10x^3\)- For \(k = 4\): \(\binom{5}{4} (1)^1 (x)^4 = 5x^4\)- For \(k = 5\): \(\binom{5}{5} (1)^0 (-x)^5 = -x^5\)

Step 5: Formulate the Expanded Expression

Combine all the terms from Step 4: \[ 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5 \]This is the expanded form of \((1-x)^5\).

Key Concepts

Understanding Binomial Coefficient

The binomial coefficient is a key part of the Binomial Theorem, used in many areas of mathematics, especially in combinatorics and algebra. The binomial coefficient, denoted as \( \binom{n}{k} \), tells us how many ways we can choose \( k \) elements from a set of \( n \) elements.
In simpler terms, it's like deciding how many different teams of \( k \) people we can form out of \( n \) candidates. It's calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) means \( n \) factorial, which is the product of all integers from 1 to \( n \). Understanding how to compute these coefficients helps with expanding expressions such as \((1-x)^5\). It tells us the weight or impact of each term as we expand a binomial.

• Example: For \( (1-x)^5 \), \( \binom{5}{2} \) represents the number of ways we can pair elements in a sequence, which is 10.

Expanding Binomials

Expanding binomials involves rewriting them as a sum of terms, using the Binomial Theorem. The theorem provides a systematic way to do this. When expanding \((1-x)^5\), you identify the pattern in the expression.
The theorem can be expressed as:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For \((1-x)^5\), set \(a = 1\), \(b = -x\), and \(n = 5\). This translates the expression using the series:

• First term: \((1)^5(-x)^0 = 1\)
• Second term: \((1)^4(-x)^1 = -5x\)
• Third term: \((1)^3(-x)^2 = 10x^2\)
• Fourth term: \((1)^2(-x)^3 = -10x^3\)
• Fifth term: \((1)^1(-x)^4 = 5x^4\)
• Sixth term: \((1)^0(-x)^5 = -x^5\)

This expansion shows that a seemingly compact expression like \((1-x)^5\) can be broken down into individual parts that can be easily understood and calculated.

The Role of Combinatorics in Binomial Expansion

Combinatorics is the branch of mathematics concerning the counting, arrangement, and combination of objects. It plays a crucial role in understanding and applying the Binomial Theorem. This relationship is visible when expanding binomials, as each term in the expansion corresponds to a specific combinatorial interpretation.
When we calculate terms using \( \binom{n}{k} \), we are essentially determining the number of combinations of selecting \( k \) items from \( n \) items, showcasing how combinatorial thinking is embedded in the procedure of binomial expansion.
Simple combinatorial ideas guide us through selecting coefficients and organizing terms, which helps in simplifying seemingly complex polynomial expressions. Thus, combinatorics allows deeper insight into patterns within algebraic expressions and is foundational in various applications within mathematics and its related fields.