Suppose that f(x)=x^3+x^2−64x−64 and g(x)=x^2−8. Find the zeros of (f◦g)(x) is/are
First, we need to find the composite function (f◦g)(x), which means applying the function f to the function g(x).
(g(x))^3 + (g(x))^2 - 64(g(x)) - 64
Substitute g(x) = x^2 - 8:
((x^2 - 8)^3) + ((x^2 - 8)^2) - 64(x^2 - 8) - 64
Expanding this function:
(x^6 - 24x^4 + 192x^2 - 512) + (x^4 - 16x^2 + 64) - 64x^2 + 512 - 64
Combine like terms:
x^6 - 23x^4 + 165x^2 + 464
Now, to find the zeros, set the function equal to zero and solve for x:
x^6 - 23x^4 + 165x^2 + 464 = 0
This is a sixth-degree polynomial equation, and finding the zeros can be complex. We can use numerical methods to approximate the zeros or plot the graph to visualize the zeros.