Does an Even Polynomial Must Have an Even Degree?

When diving into polynomials and their properties, a common question arises: if a polynomial is even, must it have an even degree? This article will explore this concept through a series of logical steps and mathematical proofs.

Defining Even Polynomials

An even polynomial is a special type of polynomial where the function remains unchanged when the input value is replaced with its negative counterpart. Mathematically, if Px is an even polynomial, it satisfies the condition:

P(-x) Px

This implies that the polynomial is symmetric about the y-axis. To better understand this, consider a polynomial expressed in its standard form:

Px a_n x^n a_{n-1} x^{n-1} … a_1 x a_0

For the polynomial to be even, the condition P(-x) Px must hold true for all x. This condition leads to the conclusion that only even-powered terms can remain, as any term with an odd power of x would change sign when x is replaced by -x. Therefore, the highest degree term, and hence the overall degree of the polynomial, must be even.

Proof by Contradiction

To further validate this, we can use a proof by contradiction. Suppose an even polynomial Px with real coefficients is of odd degree. We will show that this leads to a contradiction.

Step 1: Existence of a Real Root

According to the existence theorem, every polynomial of odd degree has at least one real root. Let’s call this root r.

Step 2: Constructing a New Polynomial

Consider the polynomial qx px / (x - r). This new polynomial qx must also be even since:

When x 0, Px 0, and hence qx 0. The condition p(-x) Px implies that q(-x) qx. Step 3: Contradiction for r 0

If r 0, then Px x(qx). This implies that Px is an odd polynomial, which contradicts our initial assumption that Px is an even polynomial.

Step 4: Contradiction for r ≠ 0

For the more complex case when r ≠ 0, we can write:

(x - r)q(x) px p(-x) -(x - r)q(-x)

Add r q(x) to both sides:

x q(x) -x q(x)

This equation holds for nonzero values of x that are not roots of q(x), leading to a contradiction since the left and right sides must have opposite signs.

Conclusion

Based on the proof by contradiction, we can conclude that an even polynomial must have an even degree. This is because all non-zero terms in the polynomial must have even degrees, ensuring that the polynomial remains unchanged under the transformation x → -x.

Summarizing, an even polynomial must have all its non-zero terms with even degrees, leading to the overall degree of the polynomial being even. This property is fundamental in understanding the behavior and properties of polynomials.