Problem 2 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In the expansion of \((1-x)^{5}\), coefficient of \(x^{5}\) will be (a) 1 (b) \(-1\) (c) 5 (d) \(-5\)

Step-by-Step Solution

Answer

The coefficient of \(x^5\) is \(-1\).

Step 1: Understand the binomial expansion form

The binomial theorem states that for any positive integer \(n\), \((a + b)^n\) can be expanded as \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our problem, the expression \((1-x)^5\) corresponds to \(a = 1\) and \(b = -x\).

Step 2: Identify the term for coefficient of \(x^5\)

To find the coefficient of \(x^5\), we need to determine the value of \(k\) in the expansion so that the term contains \((-x)^k = x^5\). This happens when \(k = 5\).

Step 3: Use the binomial coefficient formula

In our formula, the term \(\binom{n}{k} (-x)^k\) becomes \(\binom{5}{5} (-x)^5\) for our specific expression. Here \(\binom{5}{5} = 1\). Thus, the term is \(1 \times (-x)^5 = -x^5\).

Step 4: Find the coefficient of \(x^5\)

We conclude that the coefficient of \(x^5\) is found from \(-x^5\), which simply is \(-1\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used to express a polynomial in an extended form. This allows us to write expressions like \((a + b)^n\) as a series of terms. Each term in a polynomial comes from multiplying terms in the base expression and applying arithmetic rules.
For instance, using the polynomial expansion, we can break down \((1-x)^5\) into simpler terms by expanding it into \(1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5\).
The importance of polynomial expansions lies in simplifying complex calculations and making it easier to assess polynomial equations. By expanding a polynomial, we gain insights into its behavior, roots, and values for different variables.

Binomial Theorem

The binomial theorem is a crucial formula in algebra that provides a blueprint for expanding powers of binomials, which are expressions with two terms.
To apply this theorem, we use the formula: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\].
This formula helps us to systematically determine each term in the expansion without directly multiplying, which becomes harder as powers increase.

• Each term involves binomial coefficients \(\binom{n}{k}\), calculated as a combination of \(n\) items taken \(k\) at a time.
• In the expansion, coefficients and powers of each term depend on the values of \(n\) and \(k\).

Whenever you encounter a binomial to any integer power, using the binomial theorem helps you easily expand and identify the terms, like in the exercise \((1-x)^5\), producing exact coefficients and expressions for each power of \(x\).

Coefficients in Algebra

Coefficients in algebra are the numerical or constant factor in front of variables in terms of a polynomial. They are crucial in expressing polynomial terms and finding relationships between variables.
In expressions like \(ax^n\), the coefficient is \(a\). For the polynomial expansion derived from the binomial theorem, calculating coefficients becomes straightforward using binomial coefficients.

• In the binomial expansion of \((1-x)^5\), the coefficient for each term with \(x^k\) is \(\binom{5}{k}\).
• For example, the coefficient of \(x^5\) was found by identifying \(k=5\), leading to \(\binom{5}{5}(-1)^5 = -1\).
• Recognizing coefficients helps understand how much each term contributes to the polynomial's value.

Through this understanding, you can tackle algebra problems involving polynomials more easily and accurately.