Problem 19 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The expansion \(\left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5}+\left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5}\) is \(a\) polynomial of degree (a) 5 (b) 6 (c) 7 (d) 8

Step-by-Step Solution

Answer

The polynomial is of degree 7.

Step 1: Simplify the Expression

Start by examining the expression given in the problem: \( \left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5}+\left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5} \). Notice that it consists of two identical terms, each raised to the power of 5. The expression simplifies to \( 2 \times \left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5} \).

Step 2: Identify the Maximum Degree

To find the degree of the polynomial, we need to determine the maximum power of \( x \) in the expression. Consider \( x + \left(x^3 - 1\right)^{\frac{1}{2}} \) which is the base of the term raised to the 5th power. The term \( \left(x^3 - 1\right)^{\frac{1}{2}} \) has a maximum degree of \( \frac{3}{2} \) because of the cube root, making the degree of the entire base \( x + x^{\frac{3}{2}} \) equal to \( 1.5 \) or \( x^{1.5} \).

Step 3: Calculate Degree of the Expression

Raise the base \( x + x^{\frac{3}{2}} \) taken from the previous step to the 5th power. The expression would expand following binomial expansion, considering the highest term, \( x^{\frac{3}{2}} \), raised to the power 5, which gives \( (x^{\frac{3}{2}})^5 = x^{\frac{15}{2}} = x^{7.5} \). Thus, the maximum degree of any term in the expansion, and thus the degree of the polynomial, is 7.

Key Concepts

Binomial Expansion

When you need to expand an expression raised to a power, like \( \left( a + b \right)^n \), binomial expansion is key. This mathematical technique involves expressing the power expansion as a sum of terms using a formula known as the Binomial Theorem. Each term of the expansion is a binomial coefficient multiplied by powers of the original terms.
Here's a quick rundown of how it works:

• The expression \( \left( x+y \right)^n \) expands using coefficients derived from Pascal's Triangle or calculated using combinations \( \binom{n}{k} \).
• The terms start with \( x^n \) and decrease in power, while \( y \) starts from 0 and increases to \( n \).

In the context of our problem, we focus on the term \( x + (x^3 - 1)^{\frac{1}{2}} \) raised to the fifth power. Binomial expansion helps identify each resultant term's degree, ensuring we capture the term with the highest degree.

Algebra

Algebra involves the manipulation of symbols and variables to solve equations and understand equations' structures, like our given expression. Algebraic processes can simplify expressions to make calculations easier. This helps in identifying the main characteristics, like the degree of the polynomial.
In the exercise, algebra features prominently as we:

• Combine like terms or see duplicates (e.g., identifying that the original expression has identical binomials summed).
• Simplify complex expressions before further analysis (like reducing \( 2 \times \left(x + (x^3 - 1)^{\frac{1}{2}} \right)^5 \) to an easier-to-use form).

By understanding the algebraic elements, we can develop solutions more effectively and deduce properties like the polynomial's degree.

Polynomial Simplification

Finding the degree of a polynomial involves simplifying the expression to uncover the term with the highest power of \( x \). Polynomial simplification reduces the complexity, making it easier to see each component's contribution to the overall degree.
Let's dive into the problem's simplification process:

• We first notice identical parts in the expression: \( \left[x+(x^{3}-1)^{\frac{1}{2}}\right]^{5} + \left[x+(x^{3}-1)^{\frac{1}{2}}\right]^{5} \) reduces to \( 2\left[x+(x^{3}-1)^{\frac{1}{2}}\right]^{5} \).
• This simplifies our job by only having to calculate the single expression's expansion and applying the coefficient afterward.

Each simplification step brings clarity and focus to finding the polynomial's degree.

Power Functions

The core of understanding degrees in expressions like those in our problem involves power functions, which are functions of the form \( f(x) = ax^n \). Identifying these within an expression helps determine the highest degree, crucial for identifying the polynomial degree.
In this solution:

• We analyze the base expression \( x+ (x^3 - 1)^{\frac{1}{2}} \), noticing the separate terms have different degrees.
• Through expanding \( (x^{\frac{3}{2}})^5 \), the greatest degree and, therefore, the highest term in the polynomial is \( x^{7.5} \).

Handling these functions effectively helps unravel which power most influences the degree of the expression, providing the final answer to our polynomial's properties.