Problem 7 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Find the number of terms in the expansions of the following. \((2 x-3 y)^{9}\)

Step-by-Step Solution

Answer

The expansion of \\( (2x - 3y)^9 \\\) has 10 terms.

Step 1: Understanding the binomial expansion

The expansion of a binomial \( (a+b)^n \) contains \( n+1 \) terms. This is due to the Binomial Theorem, which states that the expansion of \( (a + b)^n \) is given by \( \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \).

Step 2: Identifying the values in the expression

In the expression \( (2x - 3y)^9 \), it is clear that \( 2x = a \) and \( -3y = b \). We also see that \( n = 9 \).

Step 3: Calculating the number of terms

Since the binomial expansion \( (a+b)^n \) consists of \( n+1 \) terms, substitute \( n = 9 \) into this expression to find the number of terms: \( 9 + 1 = 10 \). The expansion will include 10 terms.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that allows us to expand expressions raised to a power. When dealing with expressions like \((a+b)^n\), the binomial theorem provides a way to represent them as a sum of terms, each involving coefficients, powers of \(a\), and powers of \(b\). This theorem is expressed mathematically as:\[(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]- **Key Components:** - \(n\): The power to which the binomial is raised. - \(\binom{n}{k}\): Binomial coefficients, representing the number of ways to choose \(k\) successes from \(n\) trials. - \(a^{n-k} b^k\): The terms in the expansion where the power \(n-k\) is applied to \(a\) and \(k\) to \(b\).This expansion results in \(n+1\) terms, which is essential in calculating the number of terms in any given binomial expansion.

Number of Terms

Understanding how to determine the number of terms in a binomial expansion is crucial. For any expression of the form \((a+b)^n\), the number of terms is straightforwardly \(n+1\). This rule is derived directly from the binomial theorem, where each combination of the coefficients and powers results in a distinct term.For example, in the expression \((2x - 3y)^9\), the value of \(n\) is 9. Therefore, substituting into our formula gives:- \(n+1 = 9+1 = 10\)This means there are 10 terms in the expansion. No matter how complicated the binomial appears, applying this formula will always yield the correct number of terms in the expansion.

Algebraic Expressions

Algebraic expressions are simply mathematical phrases that include numbers, variables, and operations. These can range from simple expressions like \(x + 2\) to more complex ones like \((2x - 3y)^9\).When expressions are raised to a power, we often use the binomial theorem to expand them. This helps us write the expression as a polynomial, simplifying operations such as addition or subtraction with other polynomials. In our exercise, the expression \((2x - 3y)^9\) was expanded using the binomial theorem, clarifying how each term interacts as a part of the overall binomial expansion.It's important to recognize the basic components:- **Constants:** Numbers without variables, e.g., \(2\) in \(2x\).- **Variables:** Symbols that represent numbers, e.g., \(x\) and \(y\) in the expression.- **Coefficients:** Numbers multiplied by the variables, such as \(2\) in \(2x\) and \(-3\) in \(-3y\).Through this understanding, the complexity of algebraic expressions becomes more manageable, making it possible to work through expansions methodically.