Problem 1 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Expand \((1-2 x)^{5}\) by the binomial theorem.

Step-by-Step Solution

Answer

The expansion of \((1 - 2x)^5\) is \(1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5\).

Step 1: Understand the Formula

The binomial theorem states that \((a + b)^n\) can be expanded as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). In this exercise, we use \(a = 1\), \(b = -2x\), and \(n = 5\).

Step 2: Apply the Binomial Theorem

We will expand \((1 - 2x)^5\) by applying the binomial expansion formula. This involves calculating each term using: \( \binom{5}{k} (1)^{5-k} (-2x)^k \) for \(k = 0, 1, 2, 3, 4, 5\).

Step 3: Calculate Each Term

Compute each term by substituting the values for \(k\):- For \(k = 0\): \( \binom{5}{0}(1)^{5-0}(-2x)^0 = 1 \)- For \(k = 1\): \( \binom{5}{1}(1)^{5-1}(-2x)^1 = -10x \)- For \(k = 2\): \( \binom{5}{2}(1)^{5-2}(-2x)^2 = 40x^2 \)- For \(k = 3\): \( \binom{5}{3}(1)^{5-3}(-2x)^3 = -80x^3 \)- For \(k = 4\): \( \binom{5}{4}(1)^{5-4}(-2x)^4 = 80x^4 \)- For \(k = 5\): \( \binom{5}{5}(1)^{5-5}(-2x)^5 = -32x^5 \)

Step 4: Combine the Terms

Combine all the terms to construct the expanded expression:\((1 - 2x)^5 = 1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5\).

Key Concepts

Binomial Expansion

The process of binomial expansion involves turning expressions of the form \((a + b)^n\) into a sum of terms, each involving powers of \(a\) and \(b\). This expansion is made possible with the binomial theorem, which provides a systematic way to break down and calculate each term in the expansion using the formula:

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

In the case of \((1 - 2x)^5\), the expansion resulted in an expression with six terms, each involving powers of \(x\). Each term in the series corresponds to a different value of \(k\) ranging from 0 to 5, as demonstrated in the solution provided.
In practical applications, binomial expansion allows us to compute powers of polynomials without the need for lengthy multiplication.

Combinatorics

Combinatorics is an area of mathematics focusing on counting and arrangement. Within binomial expansion, combinatorics plays a vital role through the use of binomial coefficients, denoted as \(\binom{n}{k}\). Each binomial coefficient represents the number of ways \(k\) items can be chosen from \(n\) items, also known as "n choose k."
These coefficients can be calculated using the formula:

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

In our specific problem, these coefficients help in determining the weight each term has in the expanded polynomial. For instance, \(\binom{5}{2} = 10\), indicating the third term of \((1 - 2x)^5\) will be multiplied by 10.
This conceptual understanding of combinatorics ensures accuracy when working with larger powers and more complex expansions.

Exponentiation

Exponentiation is the mathematical operation involving numbers raised to powers. This core concept within the binomial expansion helps in breaking down each component of an expression into terms involving powers.
For a polynomial like \((1 - 2x)^5\), exponentiation is used at each step of the expansion to calculate powers of \(-2x\) for different values of \(k\). The term \((-2x)^k\) changes from one term to another as \(k\) increases, allowing us to identify the correct power to use in each part of the expanded polynomial.

• For example, when \(k = 2\), the operation \((-2x)^2\) results in \(4x^2\).
• This ensures each term in the series is accurately represented with its corresponding power.

By understanding exponentiation, we gain insights into how changes in base and power affect terms in a polynomial.

Polynomial Expansion

Polynomial expansion references the full breakdown of expressions like \((1 - 2x)^5\) into simpler components expressed as a sum of terms.
Each polynomial expansion involves transforming higher powers into sums of multiple terms that reveal individual parts by leveraging the binomial theorem.
The resulting expanded form of \((1 - 2x)^5\) is a polynomial of degree 5:

• \(1 - 10x + 40x^2 - 80x^3 + 80x^4 - 32x^5\)

This expanded polynomial presents each degree of \(x\) systematically, starting from \(x^0\) moving to \(x^5\). Such expansions not only simplify complex powers but also make further algebraic manipulation, like integration and differentiation, more manageable.

• Understanding polynomial expansion is crucial for solving mathematical equations in various fields such as physics, engineering, and economics.
• It provides a straightforward method to approach mathematical problems involving higher degree expressions.