Problem 131 - A Complete Resource Book in Mathematics for JEE Main

Question

The maximum number of positive real roots of a polynomial equation \(f(x)=0\) is the number of changes of signs from positive to negative and negative to positive in \(f(x)\). For example, consider the equation \(f(x)=x^{3}+6 x^{2}+11 x-\) \(6=0\). The signs of the various terms are: $$ +++- $$ Clearly, there is only one change of sign in the given expression. So, the given equation has at most one positive Ireal root. The maximum number of negative real roots of a polynomial equation \(f(x)=0\) is the number of changes of signs from positive to negative and negative to positive in \(f(-x)\). For example, for the equation \(f(x)=x^{4}+x^{3}+x^{2}-x-1=\) 0 , there are three changes of signs in \(f(-x)\). So, the given equation has atmost three negative real roots. If \(f(x)\) and \(f(-x)\) do not have any changes of signs, the equation \(f(x)=0\) has no real roots. Now, consider the polynomial $$ P_{n}(x)=1+2 x+3 x^{2}+\ldots+(n+1) x^{n} . $$ If \(n\) is even, the number of real roots of \(P_{n}(x)\) is (A) 0 (B) \(n\) (C) 1 (D) None of these

Step-by-Step Solution

Answer

(A) 0, as there are no real roots for \( P_n(x) \) with even \( n \).

Step 1: Analyze the Polynomial

The polynomial given is \( P_n(x) = 1 + 2x + 3x^2 + \ldots + (n+1)x^n \). This is a polynomial with positive coefficients for all powers of \( x \).

Step 2: Determine Signs of Coefficients

Since all coefficients of \( P_n(x) \) are positive, there are no changes in sign when we list them. This indicates that there are no positive real roots for \( P_n(x) = 0 \).

Step 3: Analyze the Negative Roots using \( P_n(-x) \)

Substitute \( -x \) into the polynomial: \( P_n(-x) = 1 - 2x + 3x^2 - 4x^3 + \ldots + (-1)^n(n+1)x^n \). Observe the change of signs for different powers of \( x \).

Step 4: Count Changes of Signs in \( P_n(-x) \)

Each term of \( P_n(-x) \) will alternate in sign since the powers of \( x \) change from even to odd. This will result in a change of sign between every term for an even \( n \), resulting in \( n+1 \) changes.

Step 5: Conclusion on Number of Real Roots

There are no changes of signs for \( P_n(x) \) indicating no positive real roots. However, for an even \( n \), \( P_n(-x) \) has \( n+1 \) changes of sign, indicating \( n+1 \) maximum negative real roots. Thus, there are no real roots since positive and negative roots contradict each other and cannot exist simultaneously without contradiction.

Key Concepts

Real Roots

When we talk about the real roots of a polynomial equation, we're looking for the solutions where the polynomial equals zero using real numbers. Real roots are important because they represent the actual points where the graph of the polynomial crosses the x-axis on a coordinate plane. Understanding real roots is essential for analyzing the behavior of polynomial functions and solving real-world problems.
For a polynomial, the possible number of real roots is determined by counting the sign changes in its terms. These sign changes help us visualize where the polynomial might intersect with the x-axis.

Sign Changes

Sign changes in a polynomial are a crucial part of determining its real roots. A sign change occurs when consecutive terms in a polynomial change from positive to negative or from negative to positive. Let's imagine a simple case:

• Consider the polynomial: \( f(x) = x^3 - x^2 + x - 1 \)
• The signs of the terms are: "+ - + -"
• Here, the sign changes from '+' to '-' three times.

These changes indicate the potential locations of real roots. The Descartes' Rule of Signs provides insight into how many positive or negative real roots there could be based on these changes. Therefore, identifying sign changes is a methodical way to estimate the number of real roots without solving the polynomial directly.

Negative Roots

Negative roots of a polynomial can be determined by checking the sign changes in the transformed polynomial using \( f(-x) \), which reveals potential negative solutions. By substituting \(-x\) for \(x\), we essentially flip the signs of terms with odd degrees.

• For example, if we have \( f(x) = x^4 + x^3 + x^2 - x - 1 \)
• The transformed polynomial would be \( f(-x) = x^4 - x^3 + x^2 + x - 1 \)
• The sign changes here are "+ - + + -", showing three changes.

Each of these changes indicates a possible negative real root. Using this approach enables us to systematically uncover where negative roots might occur.

Positive Roots

To find the number of positive roots in a polynomial equation, we analyze the original equation \( f(x) \) directly. Pay attention to how the signs of each term change as you move along the polynomial. Here’s how you can do it:

• Consider a polynomial such as \( f(x) = x^3 - 6x^2 + 11x - 6 \)
• The sign pattern is "+ - + -" across its terms.
• These indicate three places where the sign changes.

According to Descartes' Rule of Signs, the number of positive real roots a polynomial might have is based on these changes. More sign changes mean more possible positive real roots. By counting them, you can assess the potential for positive solutions without directly solving the polynomial.