Problem 79 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that a polynomial of an even degree has at least two real roots if it attains at least one value opposite in sign to the coefficient of its highest degree term.

Step-by-Step Solution

Answer

Given a polynomial of even degree: \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), with \(n\) even and \(a_n \neq 0\), and a value \(x_1\) such that \(P(x_1)\) has an opposite sign to \(a_n\). We know that as \(x\) approaches \(\pm\infty\), \(P(x)\) approaches \(\pm\infty\) depending on the sign of \(a_n\). So, the function must cross the x-axis at least once as it goes from the negative value to positive infinity and once more as it goes from positive infinity to the negative region where \(P(x_1)\) lies. By the Intermediate Value Theorem, there must be at least two real roots for the given polynomial.

Step 1: Define the given polynomial and value conditions

Let's consider a polynomial of even degree: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\] where \(n\) is even and \(a_n \neq 0\). Now, we are given that the polynomial has at least one value opposite in sign to the coefficient of its highest degree term, i.e., there exists an \(x_1\) such that \(P(x_1)\) has an opposite sign to \(a_n\).

Step 2: Write general behavior of an even-degree polynomial at positive and negative infinity

For an even-degree polynomial, as \(x\) approaches positive infinity (\(+\infty\)), the polynomial also approaches \(\pm\infty\), depending on the sign of the leading coefficient \(a_n\). Similarly, as \(x\) approaches negative infinity (\(-\infty\)), the polynomial also approaches \(\pm\infty\), depending on the sign of \(a_n\). So, we can write: \[\lim_{x \to +\infty}P(x) = \pm\infty\] (depending on the sign of \(a_n\)) \[\lim_{x \to -\infty}P(x) = \pm\infty\] (depending on the sign of \(a_n\))

Step 3: Use the Intermediate Value Theorem (IVT)

Now, let's assume that \(P(x_1)\) has the opposite sign to \(a_n\). Without loss of generality, let's consider \(a_n\) to be positive. Then, according to the behavior of the polynomial at infinity stated in step 2, as \(x\) approaches positive infinity, the value of \(P(x)\) must approach \(+\infty\), since \(a_n > 0\). Since there exists an \(x_1\) with \(P(x_1)\) having the opposite sign to \(a_n\), we know that \(P(x_1) < 0\). The function then must cross the x-axis at least once as it goes from this negative value to positive infinity. Now, as we approach negative infinity, the value of \(P(x)\) must also approach \(+\infty\), so the function must cross the x-axis once again as it goes from positive infinity to the negative region where \(P(x_1)\) lies. By the Intermediate Value Theorem, which states that for a continuous function in a given interval, if a positive and a negative value are attained at the endpoints of the interval, then the function must attain value 0 at least once within the interval, we can conclude that there must be at least two real roots for the given polynomial.

Key Concepts

Even-degree Polynomial

An even-degree polynomial is a polynomial function where the highest exponent of the variable is even. This means, in mathematical terms, that a polynomial of this type can be written generally as: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\]where \(n\) is an even number, and \(a_n\) is the leading coefficient which cannot be zero.

**Key Characteristics:**

• The graph of an even-degree polynomial can look like a "U" or an inverted "U" depending on the sign of \(a_n\).
• If \(a_n > 0\), both ends of the graph will rise to positive infinity.
• If \(a_n < 0\), both ends of the graph will fall to negative infinity.
• They are symmetric relative to the vertical direction, meaning what the graph does on the left side, it mirors on the right side.

Even-degree polynomials, due to their symmetry and end-behavior, naturally suggest the possibility of real roots, especially when certain conditions are met, like a sign change indicated by the exercise.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a key concept in calculus, often used in solving for roots of polynomial equations. It states that for any continuous function \(f(x)\) over an interval \([a, b]\), if \(f(a)\) and \(f(b)\) have opposite signs, then there exists at least one \(c\) in the open interval \((a, b)\) such that \(f(c) = 0\). This means the function crosses the x-axis somewhere between \(a\) and \(b\).

**What IVT Implies for Polynomials:**

• Polynomials are continuous, which means they satisfy the condition for IVT.
• If you find that a polynomial evaluates to a positive value at one point and to a negative value at another, IVT guarantees that there is at least one real root between these two points.
• This theorem is incredibly powerful for understanding the behavior of polynomial zeros (roots).In the context of our exercise, using the IVT helps ascertain that since the polynomial must achieve a positive and a negative value, there must be transitions through zero, effectively guaranteeing the existence of real roots.

Polynomial Behavior

Understanding the overall behavior of a polynomial function is crucial for predicting how many roots it might have and where these roots may lie.

**Key Behavioral Points of a Polynomial:**

• Polynomials have turning points, which are peaks and troughs where the graph changes direction.
• The degree of a polynomial gives the maximum number of real roots. For example, a fourth-degree polynomial can have up to four real roots.
• For even-degree polynomials, both ends either rise or fall together (as mentioned), which makes them especially interesting when there is a change in sign somewhere along their path.
• The effect of a sign change: When a polynomial suddenly shifts from positive to negative or vice versa, this indicates a crossing of the x-axis, hence a real root.

In our exercise, knowing the polynomial's behavior, especially its reaction to infinite limits at \( x \to \pm \infty \), and the effect of a sign change, allows us to conclude that the function must cross the x-axis at least twice, confirming the existence of at least two real roots.