Express
P(x) = x^5 + 6 x^3 - x^2 - 2x + 5
as the sum of an odd function and even function.
Podd(x)=x^5 +6x^3- 2x
Peven(x)= -x^2 + 5
To express the function P(x) as the sum of an odd function and an even function, we can use the properties of odd and even functions.
An odd function is defined as a function that satisfies the condition f(-x) = -f(x) for all x in its domain. This means that if you replace x with -x in the function, the sign of the function should change.
An even function is defined as a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if you replace x with -x in the function, the sign of the function should remain the same.
Now, let's split the given function P(x) into odd and even parts:
The odd part, P_odd(x), is obtained by substituting -x for x in P(x) and changing the sign:
P_odd(x) = -P(-x)
Substituting -x in P(x):
P_odd(x) = (-x)^5 + 6(-x)^3 - (-x)^2 - 2(-x) + 5
= -x^5 - 6x^3 - x^2 + 2x + 5
So, P_odd(x) = -x^5 - 6x^3 - x^2 + 2x + 5
The even part, P_even(x), is obtained by adding P(x) and P_odd(x) and dividing by 2:
P_even(x) = (P(x) + P_odd(x))/2
Substituting the values of P(x) and P_odd(x):
P_even(x) = (x^5 + 6x^3 - x^2 - 2x + 5 - x^5 - 6x^3 - x^2 + 2x + 5)/2
= (-x^2 + 5)
So, P_even(x) = -x^2 + 5
Therefore, the original function P(x) can be expressed as the sum of an odd function and an even function:
P(x) = P_odd(x) + P_even(x)
= (-x^5 - 6x^3 - x^2 + 2x + 5) + (-x^2 + 5)
= x^5 + 6x^3 - 2x + 5
Therefore, P(x) = x^5 + 6x^3 - 2x + 5 can be expressed as the sum of an odd function (P_odd(x) = -x^5 - 6x^3 - x^2 + 2x + 5) and an even function (P_even(x) = -x^2 + 5).