Even and Odd Polynomials
Polynomials can be classified as even, odd, or neither, based on their symmetry properties. Even polynomials are symmetric about the yaxis, 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
 Even Polynomial:
f(x) = x^2
. This quadratic function is symmetric about the yaxis.  Odd Polynomial:
f(x) = x^3
. This cubic function shows point symmetry about the origin.  Even Polynomial:
f(x) = x^4 + x^2
. Contains only even powers ofx
, showing yaxis symmetry.  Odd Polynomial:
f(x) = x^5  x^3
. Composed of odd powers ofx
, exhibiting origin symmetry.  Even Polynomial:
f(x) = 4x^4  2x^2 + 1
. Remains unchanged ifx
is replaced byx
.  Odd Polynomial:
f(x) = 3x^5 + 2x
. Changes sign but not absolute value whenx
is replaced byx
.  Even Polynomial:
f(x) = 1 + cos(x)^2
. Though not a polynomial by strict definition, it illustrates even function behavior with cosines.  Odd Polynomial:
f(x) = sin(x) * x
. Also, not a strict polynomial but shows how multiplying an odd function (sin(x)) byx
gives an odd function.  Neither Even nor Odd:
f(x) = x^3 + x^2
. This polynomial does not exhibit symmetry around the yaxis or origin.  Neither Even nor Odd:
f(x) = x^4 + x
. Similarly, lacks the symmetry to be classified as even or odd.
Packet
 Practice Solutions
 Corrective Assignments
