Problem 2 Recall that a function f is a odd if fx fx b even

Problem 2: Recall that a function f is (a) odd if f(x) = f(x) (b) even if f(x) = f(x) (c) neither even nor odd if (a) and (b) fail .

1.Show that a polynomial p(x) that contains only odd powers of x is an odd function.

2.Show that a polynomial p(x) that contains only even powers of x is an even function.

3.Show that if a polynomial p(x)contains both odd and even powers of x then it is neither an odd nor an even function.

4. Express the function

p(x) = x^5 +6x^3 x^2 2x+5 as the sum of an odd function and an even function.

Solution

Let P(x) = Ax + Bx^3 + Cx^5 + Dx^7 ...
Show that P(-x) = -P(x)

Let P(x) = A + Bx^2 + Cx^4 + Dx^6 ...
Show that P(-x) = P(x)

Let P(x) = A + Bx + Cx^2 + Dx^3 ...
Calculate P(-x).
Show that this is not the same as P(x) nor -P(x).

p(x) = x^5 +6x^3 x^2 2x+5 =  (x^5 +6x^3 2x) + (-x^2 + 5)