g(t) = t5 − 4t3 + 4t(a) Find the zeros algebraically. (Enter your answers as a comma-separated list. If there are no real zeros, enter DNE.)

To find the zeros of the function g(t) = t^5 - 4t^3 + 4t, we need to set the function equal to zero and solve for t.

0 = t^5 - 4t^3 + 4t

We can factor out a t from each term:

0 = t(t^4 - 4t^2 + 4)

Now we have a quadratic equation in the form of t^2. Let's set u = t^2. Then the equation becomes:

0 = u^2 - 4u + 4

This is a standard quadratic equation, which we can solve using the quadratic formula:

u = [-(-4) ± sqrt((-4)^2 - 414)] / (2*1) u = [4 ± sqrt(16 - 16)] / 2 u = [4 ± sqrt(0)] / 2 u = [4 ± 0] / 2 u = 2

So, t^2 = 2. To find t, we take the square root of both sides:

t = sqrt(2), -sqrt(2)

So, the zeros of the function g(t) = t^5 - 4t^3 + 4t are t = 0, sqrt(2), -sqrt(2).