Problem 8 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The total number of terms in the expansion of \((x+a)^{100}+(x-a)^{100}\) after simplification will be (a) 202 (b) 51 (c) 50 (d) none of these

Step-by-Step Solution

Answer

There are 51 terms after simplification.

Step 1: Expand using binomial theorem

\((x+a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} a^k\)
\((x-a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} (-a)^k = \sum_{k=0}^{100} \binom{100}{k} (-1)^k x^{100-k} a^k\)

Step 2: Add the expansions

When we add, terms where \(k\) is odd cancel (since \((-1)^k = -1\)), and terms where \(k\) is even double:
\((x+a)^{100} + (x-a)^{100} = 2\sum_{j=0}^{50} \binom{100}{2j} x^{100-2j} a^{2j}\)

Step 3: Count the terms

The summation runs from \(j = 0\) to \(j = 50\), giving \(50 - 0 + 1 = 51\) terms.
The answer is (b) 51.

Key Concepts

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a polynomial, using algebraic terms. This process is largely facilitated by the application of the Binomial Theorem. When expanding a binomial of the form \((x+a)^n\), it becomes a sum of terms that each involve the product of powers of \(x\) and \(a\). Every term in this expansion is a combination of these two variables, specifically following a pattern of decreasing powers of \(x\) and increasing powers of \(a\). For example, in \((x+a)^{100}\), the first term would be \(x^{100}\) and the last term would be \(a^{100}\).
Polynomial expansions are useful for computations involving large powers and can provide insights into the behavior of polynomial functions. In algebra, recognizing and working with these expanded forms allows for deeper understanding and easier manipulation of equations.

Binomial Coefficients

Binomial coefficients play a crucial role in the process of polynomial expansion. These coefficients are the numerical factors that accompany each term in a binomial expansion. They are represented by \(\binom{n}{k}\), which is read as "n choose k". This notation indicates the number of ways to choose \(k\) items from a total of \(n\) items, which is expressed mathematically by the formula:

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

Here, \(n!\) ("n factorial") represents the product of all positive integers up to \(n\). Binomial coefficients have symmetrical properties and their values can be easily found in Pascal's Triangle, a useful arrangement of numbers that shows these coefficients and their relationships.
In the context of expanding \((x+a)^{100}\) or \((x-a)^{100}\), each term includes a binomial coefficient that multiplies the variables' powers. Recognizing how these coefficients function allows for a better understanding of the expansion's structure and symmetry.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (like addition or multiplication). These expressions can range from simple ones like \(x + 2\) to more complex forms like polynomial expansions. Binomials, such as \((x+a)\) and \((x-a)\), serve as the building blocks for many algebraic expressions.
In algebra, evaluating these expressions often requires substituting values for the variables or simplifying them using known rules and theorems like the Binomial Theorem. Simplification involves reducing the complexity of expressions while maintaining their value, often making them easier to analyze or solve in equations.
When faced with problems such as expanding and simplifying \((x+a)^{100}+(x-a)^{100}\), an understanding of how algebraic expressions work, along with algebraic identities, is essential. By combining these individual expressions and simplifying them, students can achieve a final expression that is easier to interpret and use.