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March 23, 2024
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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
.
f(x) = x^2
. This quadratic function is symmetric about the yaxis.f(x) = x^3
. This cubic function shows point symmetry about the origin.f(x) = x^4 + x^2
. Contains only even powers of x
, showing yaxis symmetry.f(x) = x^5  x^3
. Composed of odd powers of x
, exhibiting origin symmetry.f(x) = 4x^4  2x^2 + 1
. Remains unchanged if x
is replaced by x
.f(x) = 3x^5 + 2x
. Changes sign but not absolute value when x
is replaced by x
.f(x) = 1 + cos(x)^2
. Though not a polynomial by strict definition, it illustrates even function behavior with cosines.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.f(x) = x^3 + x^2
. This polynomial does not exhibit symmetry around the yaxis or origin.f(x) = x^4 + x
. Similarly, lacks the symmetry to be classified as even or odd.Packet
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