Even and Odd Polynomials

Polynomials can be classified as even, odd, or neither, based on their symmetry properties. Even polynomials are symmetric about the y-axis, meaning their graph remains unchanged if x is replaced with -x. Mathematically, a polynomial f(x) is even if f(x) = f(-x) for all x. Odd polynomials exhibit point symmetry about the origin, meaning their graph rotates 180 degrees about the origin without changing. This occurs if f(-x) = -f(x) for all x.

Examples

1. Even Polynomial: f(x) = x^2. This quadratic function is symmetric about the y-axis.
2. Odd Polynomial: f(x) = x^3. This cubic function shows point symmetry about the origin.
3. Even Polynomial: f(x) = x^4 + x^2. Contains only even powers of x, showing y-axis symmetry.
4. Odd Polynomial: f(x) = x^5 - x^3. Composed of odd powers of x, exhibiting origin symmetry.
5. Even Polynomial: f(x) = 4x^4 - 2x^2 + 1. Remains unchanged if x is replaced by -x.
6. Odd Polynomial: f(x) = -3x^5 + 2x. Changes sign but not absolute value when x is replaced by -x.
7. Even Polynomial: f(x) = 1 + cos(x)^2. Though not a polynomial by strict definition, it illustrates even function behavior with cosines.
8. Odd Polynomial: f(x) = sin(x) * x. Also, not a strict polynomial but shows how multiplying an odd function (sin(x)) by x gives an odd function.
9. Neither Even nor Odd: f(x) = x^3 + x^2. This polynomial does not exhibit symmetry around the y-axis or origin.
10. Neither Even nor Odd: f(x) = x^4 + x. Similarly, lacks the symmetry to be classified as even or odd.