Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x=0\) has a positive root \(x=\alpha\), then the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots+a_{1}=0\) has a positive root, which is (A) smaller than \(\alpha\) (B) greater than \(\alpha\) (C) equal to \(\alpha\) (D) greater than or equal to \(\alpha\)

Step-by-Step Solution

Answer

The root is smaller than \(\alpha\) (Option A).

Step 1: Understand the Given Equations

We have two polynomial equations. The original equation is \(a_{n} x^{n}+a_{n-1} x^{n-1} + \ldots + a_{1} x = 0\) which is known to have a positive root \(x = \alpha\). The second equation is \(n a_{n} x^{n-1} + (n-1) a_{n-1} x^{n-2} + \ldots + a_{1} = 0\). Our task is to determine the relation between the positive roots of these two equations.

Step 2: Consider the Derivative of the Polynomial

The second polynomial given is actually the derivative of the first polynomial with respect to \(x\). Taking the derivative of \(a_{n} x^{n}+a_{n-1} x^{n-1} + \ldots + a_{1} x = 0\), we obtain \(n a_{n} x^{n-1} + (n-1) a_{n-1} x^{n-2} + \ldots + a_{1} = 0\). This is the derivative equation.

Step 3: Use Rolle's Theorem

Rolle's Theorem states that if a continuous function has two points where it is zero, there must be at least one point between them where the derivative is zero if the function is differentiable on that interval. Since \(x = \alpha\) is a root of the original function, and assuming it's a simple root, the derivative will typically cross zero before or after \(\alpha\), indicating a positive root for the derivative polynomial.

Step 4: Apply Rolle's Theorem to the Problem

Because \(x = \alpha\) is a root of the original polynomial, and assuming the root is simple (where the function changes sign and thus crosses 0), the derivative (which is our second equation) must have a root between any two roots of the original polynomial. If \(\alpha\) is the largest root of the original equation, the root of the derivative equation must be less than \(\alpha\).

Step 5: Conclusion

Based on our analysis using Rolle's Theorem, we conclude that the second equation has a positive root smaller than \(\alpha\). This suggests option (A) is correct: the root of the derivative equation is smaller than \(\alpha\).

Key Concepts

Polynomial Equation

A polynomial equation is an expression consisting of variables raised to various powers, multiplied by coefficients, and summed to produce a zero value.
For example, in the polynomial equation given in the exercise, \(a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{1} x = 0\), we see:

• \(a_n, a_{n-1}, \ldots, a_1\) are coefficients.
• \(x\) is the variable.
• \(n, n-1, \ldots, 1\) are the exponents.

Each term is a combination of a coefficient, variable, and exponent. Polynomials can take various forms depending on the degree \(n\), which is the highest exponent in the expression.
Solving polynomial equations generally involves finding the values for \(x\) (known as roots) that satisfy the equation, meaning they make the equation equal to zero.

Derivative

The derivative of a function represents the rate at which the function's value changes as its input changes.
This concept is critical in calculus and was used in the exercise to transition from the original polynomial to its derivative.
For the given polynomial \(a_{n} x^{n}+a_{n-1} x^{n-1} + \ldots + a_{1} x = 0\), the derivative is calculated as:

• Multiply each term by its exponent.
• Reduce the exponent by one.

Thus, the derivative becomes \(n a_{n} x^{n-1} + (n-1) a_{n-1} x^{n-2} + \ldots + a_{1} = 0\). This new equation indicates how the polynomial's rate of change varies with \(x\).
In our exercise, finding a root of the derivative helps identify where the slope of the original polynomial crosses zero.

Rolle's Theorem

Rolle's Theorem is a special case of the mean value theorem in calculus. It states that if a function is continuous on a closed interval and differentiable on its open interval, and has equal values at the endpoints, then there exists at least one c in the interval where the derivative is zero.
This theorem was pivotal in the step-by-step solution because:

• It helps identify where a derivative of a function equals zero between two roots of the function.
• In our case, since \(x = \alpha\) is a root of the original polynomial, the derivative (second polynomial) must have a root between any two roots of the first equation.

Thus, Rolle's Theorem not only guides us to understand the presence of roots in derivative equations but crucially aids in predicting their location relative to existing roots.

Roots of Equations

Roots of equations signify the values of the variable that make the equation true, meaning the left side equals zero.
In polynomial equations, these roots are the solutions that satisfy the equation. In our exercise, the focus was on how the roots of a polynomial relate to the roots of its derivative.
Key points about polynomial roots:

• Roots can be real or complex numbers.
• In this particular exercise, the root \(x = \alpha\) was known to be positive.
• The analysis using Rolle's Theorem indicated the roots of the derivative equation are smaller but real, and positive if \(\alpha\) is positive.

Identifying roots provides insights into the behavior of a polynomial, such as where it crosses the x-axis, which is critical for graphing and solving these equations.