Problem 26 - Algebra and Trigonometry

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: Identify the Binomial Expansion Formula

The Binomial Theorem states that for any positive integer \( n \), \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case, we have \(a = 1\), \(b = -x\), and \(n = 5\).

Step 2: Setup the Binomial Coefficients

We need to calculate the binomial coefficients \(\binom{5}{k}\) for \(k = 0\) to \(k = 5\). The formula for binomial coefficients is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 3: Calculate Each Term of the Expansion

For each \(k\) from 0 to 5, calculate \(\binom{5}{k} (1)^{5-k} (-x)^k\) and simplify:- 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 4: Write the Final Expanded Expression

Combine all the terms to form the full expansion: \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\).

Key Concepts

Binomial Expansion

Binomial expansion uses the Binomial Theorem to express a power of a binomial, like \((a + b)^n\), as a sum of terms. Each term involves the original variables raised to successive powers and multiplied by a binomial coefficient.
Binomial expansions simplify complex algebraic expressions by breaking them down into sums.

For the expression \((1-x)^5\), the Binomial Theorem tells us:

• We start with the binomial formula \((1-x)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-x)^k\).
• This expansion results in multiple terms, specifically six terms (from \(k=0\) to \(k=5\)).
• Each term is simplified individually and then combined into a polynomial expression.

This method not only shows the power of rewriting expressions efficiently but also illuminates the concept of how algebraic tools simplify intricate mathematical problems.

Binomial Coefficients

Binomial coefficients are central to the binomial expansion process. They are the numbers appearing in the expansion of binomial powers and are represented by \(\binom{n}{k}\).
These coefficients are computed using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• The symbol \(n!\) (n factorial) represents the product of all positive integers up to \(n\).
• For instance, to find \(\binom{5}{3}\), calculate \(\frac{5!}{3!2!} = 10\).

In the expansion of \((1-x)^5\), the coefficients are calculated for each term. These values provide weights for each component of the expansion vividly illustrating the pattern's symmetry and balance. Understanding these coefficients allows us to derive each term's contribution to the entire polynomial accurately.

Polynomial Expansion

Polynomial expansion via the Binomial Theorem transforms a simple binomial raised to a power into a polynomial composed of several terms. Each term is a product of a power of the first variable, a binomial coefficient, and a power of the second variable.
For \((1-x)^5\), this leads to expanding it into:

• \(1\) for \(k=0\)
• \(-5x\) for \(k=1\)
• \(10x^2\) for \(k=2\)
• \(-10x^3\) for \(k=3\)
• \(5x^4\) for \(k=4\)
• \(-x^5\) for \(k=5\)

These individual pieces fit together like puzzle parts to form a complete polynomial. This approach not only facilitates handling of complex expressions but also aids in understanding algebraic structures and their potential expansions.

Algebraic Expressions

Algebraic expressions include variables, constants, and operators like addition and multiplication. The binomial expansion serves as a powerful tool for manipulating these expressions by breaking them down into simpler, manageable parts.

Consider the expression \((1-x)^5\):

• It initially looks simple, but its power signifies a more complex structure upon expansion.
• Through binomial expansion, it is transformed into a series of terms (a polynomial), making it easier to analyze or compute substitutions.
• The resulting expression \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\) both encapsulates the original expression's essence and offers a straightforward form for further calculations.

Algebraic manipulation via such expansions underscores the elegant depth of mathematics, showing how seemingly basic formulas conceal vast possibilities for simplification and insight.