Explain algebraically why a polynomial that is
an odd function is no longer an odd function
when a nonzero constant is added
An odd function is a function that satisfies the property f(-x) = -f(x) for all x in its domain.
Let's say we have an odd function given by f(x). Adding a nonzero constant c to this function results in a new function g(x) = f(x) + c.
To show that g(x) is no longer an odd function, we need to determine g(-x).
g(-x) = f(-x) + c
Since f(x) is an odd function, we know that f(-x) = -f(x).
Therefore, g(-x) = -f(x) + c
This is not equal to -g(x) because g(x) = f(x) + c, so -g(x) = -f(x) - c.
Since g(-x) is not equal to -g(x), adding a nonzero constant to an odd function results in a function that is not odd.