Problem 22 - Algebra and Trigonometry

Question

\(21-24\) 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: Understanding the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a+b)^n\). According to the theorem, \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Identify the Components

For the expression \((1-x)^5\), identify \(a = 1\), \(b = -x\), and \(n = 5\). We need to expand it using these values with the binomial theorem.

Step 3: Apply the Binomial Theorem

Substitute \(a = 1\), \(b = -x\) and \(n = 5\) into the binomial expansion formula: \[(1-x)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-x)^k\]

Step 4: Calculate Binomial Coefficients

Calculate each binomial coefficient \(\binom{5}{k}\) for \(k=0,1,2,3,4,5\). The coefficients are: - \(\binom{5}{0} = 1\) - \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\)- \(\binom{5}{3} = 10\)- \(\binom{5}{4} = 5\)- \(\binom{5}{5} = 1\)

Step 5: Expand Each Term of the Series

Use the binomial coefficients to expand each term: - For \(k=0\): \(1\cdot (1)^5 \cdot (-x)^0 = 1\)- For \(k=1\): \(5 \cdot (1)^4 \cdot (-x)^1 = -5x\)- For \(k=2\): \(10 \cdot (1)^3 \cdot (-x)^2 = 10x^2\)- For \(k=3\): \(10 \cdot (1)^2 \cdot (-x)^3 = -10x^3\)- For \(k=4\): \(5 \cdot (1)^1 \cdot (-x)^4 = 5x^4\)- For \(k=5\): \(1 \cdot (1)^0 \cdot (-x)^5 = -x^5\)

Step 6: Combine the Terms

Add all the terms from the expansion to get the final expression: \((1-x)^5 = 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\)

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we're referring to the process of expanding an expression elevated to a power, where the binomial consists of two terms. For instance, in the expression \((a + b)^n\), 'a' and 'b' are the two terms that are expanded through repeated multiplication. By using the formula derived from the Binomial Theorem, the task of expanding such expressions becomes systematic and less cumbersome.The beauty of binomial expansion is that it simplifies our computations, allowing us to express a binomial raised to a power as a sum of terms involving each component of the binomial. In our example problem, \((1 - x)^5\) is expanded into a full polynomial with several terms. This systematic approach is incredibly useful in algebra when dealing with any similar type of expression.

Binomial Coefficients

The numbers that form the backbone of any binomial expansion are the binomial coefficients. These coefficients appear in the formula of the Binomial Theorem and are expressed as \(\binom{n}{k}\), representing 'n choose k'. They are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]These coefficients give us the number of ways to choose 'k' successes in 'n' trials and are equivalent to the numbers found in Pascal's Triangle. For the expression \((1-x)^5\), we calculated binomial coefficients \(\binom{5}{0}\) to \(\binom{5}{5}\). Each of these values is used to determine the contribution of each term in the expansion, reflecting how important these coefficients are in constructing the polynomial.

Polynomial Expansion

The result of applying the Binomial Theorem is often a polynomial expansion, which breaks down a power of a binomial into multiple terms. This form of expansion translates the single expression \((a + b)^n\) into a sum of multiple algebraic expressions.Polynomials are expressions consisting of variables, coefficients, and exponents. In the expansion \((1-x)^5\), we find a comprehensive polynomial:

• \(1\)
• -5x
• 10x^2
• -10x^3
• 5x^4
• -x^5

Each term results from multiplying binomial coefficients by the respective powers of 'a' and 'b'. Thus, a polynomial expansion can transform complex expressions into simpler, more manageable components.

Algebraic Expressions

Algebraic expressions form the fundamental language of algebra, composed of terms involving numbers, variables, and arithmetic operations. Understanding these expressions is crucial for tasks including binomial expansions.In our expansion of \((1-x)^5\), each term is an algebraic expression—a product of constants (binomial coefficients) and powers of 'x'. These expressions provide a way to translate a mathematical statement into a format that is easier to manipulate and understand.The process of expanding and simplifying these expressions involves careful manipulation, applying the Binomial Theorem, calculating coefficients, and correctly adjusting signs based on binomial terms. Mastering this process allows students to confidently handle a broader range of algebraic problems.