Problem 175 - A Complete Resource Book in Mathematics for JEE Main

Question

How many real solutions does the equation \(x^{7}+14 x^{5}\) \(+16 x^{3}+30 x-560=0\) have? \(\quad\) [2008] (A) 7 (B) 1 (C) 3 (D) 5

Step-by-Step Solution

Answer

The equation has 5 real roots.

Step 1: Identifying the Given Polynomial

The equation given is a polynomial equation: \(x^7 + 14x^5 + 16x^3 + 30x - 560 = 0\). This is a seventh-degree polynomial because the highest power of \(x\) is 7. A polynomial of degree 7 can have at most 7 real roots.

Step 2: Applying Descartes' Rule of Signs

Descartes' Rule of Signs helps determine the number of positive and negative real roots. For positive roots, consider the changes in sign in the polynomial: \(+ + + + -\), resulting in only 1 sign change, suggesting at most 1 positive root.For negative roots, substitute \(x\) with \(-x\) to obtain: \(-x^7 + 14x^5 - 16x^3 + 30x - 560\), which has the signs \(- + - + -\). This shows 4 sign changes, suggesting up to 4 negative roots.

Step 3: Using Intermediate Value Theorem

To confirm the existence of real roots, check using the Intermediate Value Theorem by inspecting values at different points. Substituting \(x = 0\), we get \(-560\). For large values like \(x = 10\), calculations show a positive value due to the \(x^7\) term. This indicates a root because the polynomial changes from negative to positive values.

Step 4: Analyzing Root Multiplicity and Further Exploration

Detect complex conjugate pairs, if necessary, and confirm simple root listings. With 1 sign change for positive roots and up to 4 negative roots, explore calculation-based reductions, like reducing the equation or plotting. This exploration indicates 1 positive real root confirmed and a possibility of further real negative roots based on decreases from degree.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful tool in determining how many positive and negative real solutions a polynomial equation might have. By analyzing the changes in sign of the coefficients of the polynomial, you can predict the nature of its roots.
For positive roots:

• Count the number of times the sign of the coefficients change in the polynomial as it is written with positive terms of the variable. Each change represents a possible positive real solution.
• In the polynomial equation \(x^7 + 14x^5 + 16x^3 + 30x - 560 = 0\), the sequence of signs is "+ + + + -". This shows one sign change, suggesting at most one positive root.

For negative roots:

• Replace \(x\) with \(-x\) in the polynomial and count the sign changes again. Each change represents a possible negative real solution.
• In this scenario, substituting \(-x\) yields the polynomial \(-x^7 + 14x^5 - 16x^3 + 30x - 560\), which has a sign change pattern of "- + - + -", meaning four sign changes. This indicates up to four negative real roots could exist.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus used to establish the existence of a real root within an interval where a continuous function takes on both positive and negative values.
To apply the IVT to our polynomial \(x^7 + 14x^5 + 16x^3 + 30x - 560 = 0\):

• First, evaluate the function at various points to observe changes in sign. If the function changes from negative to positive or vice versa between two points, a real root lies within that interval.
• For example, at \(x = 0\), the polynomial evaluates to \(-560\) (a negative number). Calculate for a large positive value such as \(x = 10\), where the dominant \(x^7\) term makes the result large and positive.
  This change from negative to positive indicates there is at least one real root in this interval according to the IVT.

Seventh-degree Polynomial

A seventh-degree polynomial is a polynomial where the highest degree term is raised to the seventh power. These types of polynomials have several important properties that can help in analyzing their real roots.
Key characteristics include:

• The maximum number of real roots is equal to its degree, meaning up to 7 real solutions in this scenario.
• With multiple degrees like this one, the roots can be a blend of real and complex numbers. Complex roots, however, always come in conjugate pairs, so if there are complex roots, they will reduce the number of real roots from the maximum possible.
• A seventh-degree polynomial like \(x^7 + 14x^5 + 16x^3 + 30x - 560\) may require additional methods, such as graphing or numerical techniques, to ascertain the exact nature and number of its roots.

Understanding these features helps in breaking down how such a polynomial behaves across different values and why concepts like Descartes' Rule of Signs and the Intermediate Value Theorem are so crucial for solving equations like this.