Problem 9 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(p\) and \(q\) be positive, then the coefficient of \(x^{p}\) and \(x^{7}\) in the expansion of \((1+x)^{p+a}\) will be (a) equal (b) equal in magnitude but opposite in sign (c) reciprocal to each other (d) none of these

Step-by-Step Solution

Answer

The coefficients are not guaranteed to be equal, opposite, or reciprocal without further conditions, so the answer is (d) none of these.

Step 1: Understand the Binomial Theorem

The problem involves the expansion of \((1 + x)^{p+a}\) using the binomial theorem, which states that for any non-negative integer \(n\), \((1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\). Here, \(\binom{n}{k}\) is known as the binomial coefficient.

Step 2: Identify the Coefficient of \(x^p\)

The term containing \(x^p\) in the expansion \((1 + x)^{p+a}\) is represented by \(\binom{p+a}{p} x^p\). Thus, the coefficient of \(x^p\) is \(\binom{p+a}{p}\).

Step 3: Identify the Coefficient of \(x^7\)

Similarly, the term containing \(x^7\) in the expansion is represented by \(\binom{p+a}{7} x^7\). Therefore, the coefficient of \(x^7\) is \(\binom{p+a}{7}\).

Step 4: Compare the Coefficients

For option (a) to be true, \(\binom{p+a}{p}\) must equal \(\binom{p+a}{7}\). Given the properties of binomial coefficients, this is only possible when \(p = 7\). If \(p\) and \(q\) are arbitrary positive integers, there is no promise of equality for all cases. Therefore, the coefficients are typically not equal or reciprocal without additional specific conditions.

Key Concepts

Understanding Binomial Coefficients

Binomial coefficients are essential tools in algebra, particularly when dealing with polynomials. They appear in the expansion of binomials, which are algebraic expressions of the form \((a + b)^n\). Binomial coefficients are represented as \(\binom{n}{k}\), where \(n\) and \(k\) are non-negative integers. This notation is read as "n choose k" and represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order.

• The expression can be calculated using factorial notation: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• They are symmetric, meaning \(\binom{n}{k} = \binom{n}{n-k}\).
• In the binomial theorem, they represent the coefficients that accompany the terms \(x^k\) in the expansion.

Understanding binomial coefficients helps us find specific terms in polynomial expansions easily.

The Process of Polynomial Expansion

Polynomial expansion involves expressing a binomial raised to a power, say \((a + b)^n\), as a sum of terms. Each term in the expansion is formed by multiplying the binomial coefficient by appropriate powers of the variables. According to the binomial theorem, \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k.\]

• Every term \((\binom{n}{k} a^{n-k} b^k)\) represents a specific position in the expansion.
• The process involves determining the coefficient for each pair of powers of \(a\) and \(b\).
• This method applies to any non-negative integer \(n\), making it universally applicable in algebra.

Polynomial expansion is a powerful method for simplifying expressions and solving equations involving powers of binomials.

Exploring Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operators. They can take various forms, such as linear, quadratic, or more complex forms involving powers and roots. Binomial expressions, like \((1 + x)^{p+a}\), are a specific type of algebraic expression useful in various applications.

• An algebraic expression might consist of terms that include constants (numbers) and coefficients (numbers multiplying variables).
• The variables can be raised to any power, described by non-negative integers.
• Such expressions are foundational in formulating equations and inequalities used throughout algebra.

Grasping the structure and properties of algebraic expressions aids in tackling problems in calculus, statistics, and beyond. In the context of the binomial theorem, recognizing how terms like \((1 + x)^{p+a}\) can be expanded helps simplify complex equations.