Problem 175 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

The equation has 1 real solution.

Step 1: Identify the equation

We are given the equation \(x^{7}+14x^{5}+16x^{3}+30x-560=0\). Our task is to determine how many real solutions this equation has.

Step 2: Consider the degree of the polynomial

The polynomial \(x^{7}+14x^{5}+16x^{3}+30x-560\) is a degree 7 polynomial because the highest power of \(x\) is 7. A degree \(n\) polynomial can have up to \(n\) real solutions.

Step 3: Use Descartes' Rule of Signs for positive roots

Descartes' Rule of Signs can help determine the number of positive real roots. Count the number of sign changes in the coefficients of \(x^{7}, 14x^{5}, 16x^{3}, 30x, -560\). The sequence is \(+, +, +, +, -\). There is one sign change (from + to -), indicating one positive real root.

Step 4: Use Descartes' Rule of Signs for negative roots

Replace \(x\) with \(-x\) in the equation to determine the number of negative roots: \((-x)^{7}+14(-x)^{5}+16(-x)^{3}+30(-x)-560\).This becomes \(-x^{7}-14x^{5}-16x^{3}-30x-560\).Check for sign changes: \(-, -, -, -, -\). There are no sign changes, indicating no negative real roots.

Step 5: Conclude the number of real roots

From Steps 3 and 4, we identified that there is 1 positive real root and no negative real roots. Therefore, the polynomial has only 1 real solution.

Key Concepts

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful method for estimating the number of real roots in a polynomial equation, focusing specifically on positive and negative real roots. It examines how the signs of a polynomial's coefficients change.

For a polynomial written in standard form, with terms ordered by descending powers of the variable, this rule can predict:

• Positive real roots by counting the sign changes in the sequence of its coefficients. Each sign change can suggest one or more positive roots.
• Negative real roots by substituting \(x\) with \(-x\) and then counting the sign changes in the new sequence of coefficients.

In the given polynomial equation, \(x^{7} + 14x^{5} + 16x^{3} + 30x - 560\), we identify one positive real root due to a single sign change from "+" to "−". For negative roots, substituting \(x\) with \(-x\) leads to no sign changes in the coefficients. Thus, we infer that there are zero negative real roots.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable \(x\) that appears in the equation, provided the coefficient of that term is not zero.

In our problem, the polynomial is \(x^{7} + 14x^{5} + 16x^{3} + 30x - 560\). The term \(x^{7}\) shows that the polynomial has a degree of 7 since this is the highest power. Understanding the degree is key because it provides information about:

• The maximum number of roots the polynomial can have. In this case, up to 7 real roots.
• The shape of the graph of the polynomial. A 7th-degree polynomial can have complex behaviors such as multiple turning points.

Even though the degree indicates a possibility of 7 real solutions, Descartes’ Rule of Signs refines this estimate by focusing on actual real roots based on sign changes.

Positive and Negative Roots

The terms 'positive roots' and 'negative roots' refer to the real solutions of a polynomial where the roots (solutions) are either greater than zero or less than zero.

For finding positive roots, follow these steps:

• Identify the sign changes in the polynomial's coefficients as they are written.
• Each sign shift suggests one or more positive real roots.

In our polynomial, the sequence \(+, +, +, +, -\) has one sign change, indicating a single positive real root.

To find negative roots:

• Substitute \(x\) with \(-x\).
• Count sign changes in the new equation.

After substitution in this equation, the sequence becomes all negative \(-, -, -, -, -\), with no sign changes observed. Hence, no negative real roots exist.

Understanding the number of positive and negative roots is crucial not only for identifying possible solutions but also for determining the behavior and interaction of polynomial functions with the x-axis.