Problem 26 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\((1-x)^5 = 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5.\)

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here \(a = 1\), \(b = -x\), and \(n = 5\). We will use this theorem to expand \((1-x)^5\).

Step 2: Calculate the Binomial Coefficients

The binomial coefficients are given by \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) for each term, where \(k\) ranges from 0 to \(n\). For \(n=5\), we calculate: - \(\binom{5}{0} = 1\) - \(\binom{5}{1} = 5\) - \(\binom{5}{2} = 10\) - \(\binom{5}{3} = 10\) - \(\binom{5}{4} = 5\) - \(\binom{5}{5} = 1\).

Step 3: Apply the Binomial Theorem

Expand \((1-x)^5\) using the calculated coefficients. Substitute \(a = 1\) and \(b = -x\): \[ (1-x)^5 = \binom{5}{0} 1^5 (-x)^0 + \binom{5}{1} 1^4 (-x)^1 + \binom{5}{2} 1^3 (-x)^2 + \binom{5}{3} 1^2 (-x)^3 + \binom{5}{4} 1^1 (-x)^4 + \binom{5}{5} 1^0 (-x)^5 \] This simplifies to: \[ 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5 \].

Step 4: Summarize the Expansion

The expanded form of \((1-x)^5\) using the Binomial Theorem is: \[ (1-x)^5 = 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5 \].

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are essential for expanding expressions using the Binomial Theorem. These coefficients, often represented by \( \binom{n}{k} \), help determine the weight of each term in the expansion. To compute a binomial coefficient, you use the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
• The symbol \( ! \) denotes factorial, which means you multiply a series of descending natural numbers. For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Binomial coefficients are symmetric, meaning \( \binom{n}{k} = \binom{n}{n-k} \).

If you expand \((1-x)^5\), each coefficient supports a specific term. Therefore, these coefficients enhance the structure and symmetry of binomial expansions, making complex algebra simpler. They ensure that terms are accurately set based on how many times each component appears in the product.

Expanding Binomial Expressions

Expansion using the Binomial Theorem gives a stretched out version of expressions like \((a - b)^n\). This process explains why your original expression grows into many others. The formula for expansion is:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Here, \(n\) is the power, and the coefficients \(\binom{n}{k}\) detail how each term in the expansion is weighted.

For \((1-x)^5\), you substitute \(a = 1\), \(b = -x\), and \(n = 5\). The expansion results in a sequence of terms each being seamlessly constructed out of multiplying different powers of \(1\) and \(-x\). You get each sequence term by multiplying the corresponding binomial coefficient by powers of \(1\) and \(-x\), then adding all these products together. This explains the expansion to: \[1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\].

Importance of Polynomials in Binomial Expansion

Polynomials are algebraic expressions consisting of variables raised to non-negative integer powers combined using addition, subtraction, and multiplication. They become handy while dealing with expansions, like \((1 - x)^5 \). Why? Because after expanding, you obtain a polynomial. In this case, \[1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\].Here's why polynomials are crucial:

• Polynomials are simple models that can approximate more complex functions.
• They simplify calculations, as each term is distinct and manageable.
• When expanded, polynomials represent specific values' behavior around the variable used, effectively showing their rate and nature of change.

Expanding a binomial into a polynomial provides clear insights into how each part contributes to the overall structure. It highlights the stability and predictability of polynomial forms in algebra.