Detrmine algebraically whether the function is even, odd, or neither g (x) = x^7+8x^5-x^3+6x

(4x2 - 2x) - (-5x2 - 8x)

= 4x2 - 2x + 5x2 + 8x.

= 4x2 + 5x2 - 2x + 8x.

= 9x2 + 6x.

= 3x (3x + 2).

Answer: 3x (3x + 2)

Short Answer: Because all the powers are odd numbers, then the function will be odd.

This problem does not have so much to do with odd powers as it does with minus signs.

h (x) = x^7 gives x^7 when x is put in the x value

h (-x) = (-x) ^7 does not give the same thing. Seven minus signs have one unpaired minus sign left over so g (x) = - x^7 That is different than x^7.

The definition says that the entire function will be minus because all of the powers are odd. And the entire function of g (x) is odd.