Problem 95 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(q \neq 0\) and the equation \(x^{3}+p x^{2}+q=0\) has a root of multiplicity 2, then \(p\) and \(q\) are connected by (A) \(p^{2}+2 q=0\) (B) \(p^{2}-2 q=0\) (C) \(4 p^{3}+27 q+1=0\) (D) \(4 p^{3}+27 q=0\)

Step-by-Step Solution

Answer

Option (D) \(4p^3 + 27q = 0\).

Step 1: Understand the Problem

The problem gives us a cubic equation \(x^3 + px^2 + q = 0\) and says there's a root with multiplicity 2. We need to find the relationship between \(p\) and \(q\).

Step 2: Root Multiplicity Conditions

If \(x = a\) is a root with multiplicity 2, the derivative of the function must also be zero at \(x = a\). Let \(f(x) = x^3 + px^2 + q\), then the derivative \(f'(x) = 3x^2 + 2px\).

Step 3: Set Up the System of Equations

For \(x = a\) to be a root of multiplicity 2, we have:1. \(a^3 + pa^2 + q = 0\) (since \(a\) is a root of the equation)2. \(3a^2 + 2pa = 0\) (since \(a\) is also a root of the derivative).

Step 4: Simplify the System

To solve the derivative equation \(3a^2 + 2pa = 0\), factor out \(a\): \[ a(3a + 2p) = 0 \]This gives two possible solutions: \(a = 0\) or \(3a + 2p = 0\).

Step 5: Consider Case when \(a = 0\)

Substituting \(a = 0\) into the first equation \(a^3 + pa^2 + q = 0\) simplifies to \(q = 0\). But since \(q eq 0\), \(a eq 0\). Proceed with the second case.

Step 6: Solve for \(a\) Using \(3a + 2p = 0\)

Solving \(3a + 2p = 0\) gives \(a = -\frac{2p}{3}\). Substitute \(a = -\frac{2p}{3}\) into the cubic equation \(a^3 + pa^2 + q = 0\):\[\left(-\frac{2p}{3}\right)^3 + p\left(-\frac{2p}{3}\right)^2 + q = 0\]

Step 7: Simplify the Substituted Equation

Calculate each term:- \(\left(-\frac{2p}{3}\right)^3 = -\frac{8p^3}{27}\)- \(p\left(-\frac{2p}{3}\right)^2 = \frac{4p^3}{9}\)Combine them:\[-\frac{8p^3}{27} + \frac{4p^3}{9} + q = 0\]

Step 8: Solve for \(q\) in Terms of \(p\)

Multiply the entire equation by 27 to clear the fractions:\[-8p^3 + 12p^3 + 27q = 0\]Simplify:\[4p^3 + 27q = 0\]

Step 9: Identify the Correct Option

The derived relationship \(4p^3 + 27q = 0\) matches option (D).

Key Concepts

Polynomial Root Multiplicity

In mathematics, a polynomial root is said to have a multiplicity of two if a root appears twice among the polynomial's solutions. In simpler terms, if a polynomial equation like \( f(x) = 0 \) has a root \( x = a \) that satisfies the equation twice, then \( a \) is a root with a multiplicity of two. When tackling problems involving root multiplicity, it’s important to understand that not all roots have to be unique. Some roots can repeat, which affects how the polynomial behaves. Roots with higher multiplicities tend to "flatten" the graph of the polynomial at that root, making it a point where the graph "bounces off" rather than simply crossing.

• Graphical Understanding: A root with multiplicity 2 causes the polynomial graph to merely touch the x-axis at that point but not cross it.
• Algebraic Expression: For a root \( x = a \) of multiplicity 2, \( (x-a)^2 \) will be a factor of the polynomial. This implies that both \( f(a) = 0 \) and the first derivative, \( f'(a) = 0 \), must also hold true.

Recognizing and understanding root multiplicity is an integral part of solving and simplifying polynomial equations.

Cubic Equation Derivative

The derivative of a cubic equation plays a crucial role, especially when analyzing root multiplicity. Given a general cubic equation \( f(x) = x^3 + px^2 + q \), the derivative provides information about the slope or rate of change at any given point on the curve. Calculating the derivative requires applying basic calculus rules.

• Derivative Calculation: The derivative of the cubic function \( f(x) = x^3 + px^2 + q \) is \( f'(x) = 3x^2 + 2px \). This follows the power rule for differentiation, where the exponent of each term is brought down as a coefficient and reduced by one.
• Analyzing the Derivative: Knowing the derivative \( f'(x) \) is essential to determine where the function's slope is zero, which is a key criteria for root multiplicity.

Understanding the derivative helps solve for critical points where the polynomial may have roots with multiplicity greater than one, which is key to many algebraic problems.

Root Multiplicity Conditions

The conditions for root multiplicity are vital in identifying the relationship between the coefficients of a polynomial. Specifically, if a root of an equation like \( x^3 + px^2 + q = 0 \) has a multiplicity of 2, certain criteria must be satisfied. Here is a detailed look at these conditions:

• Condition of Root:  If \( x = a \) is a root, then \( f(a) = a^3 + pa^2 + q = 0 \).
• Condition on the Derivative: For the root to have a multiplicity of 2, not only \( f(a) \) should be zero, but also \( f'(a) = 0 \). This means \( 3a^2 + 2pa = 0 \), which translates to having a slope that equals zero at that point.
• Solving the System: The combined conditions \( f(a) = 0 \) and \( f'(a) = 0 \) result in a set of equations making it possible to solve for complex relationships between \( p \) and \( q \). For instance, in our problem, solving these equations yields \( 4p^3 + 27q = 0 \).

Recognizing and using these conditions helps in determining the complex relationship between the polynomial's coefficients and its roots. Proper understanding of these concepts is crucial for correct polynomial analysis and solutions.