iven the polynomial function P defined by 𝑃(𝑡)=3𝑡3−243𝑡, what are its zeros?

To find the zeros of the polynomial function P(t) = 3t^3 - 243t, we need to set the function equal to zero and solve for t.

0 = 3t^3 - 243t

First, we can factor out a common factor of 3t:

0 = 3t(t^2 - 81)

Then, we can set each factor equal to zero and solve for t:

3t = 0 --> t = 0

t^2 - 81 = 0

To solve t^2 - 81 = 0, we can use the difference of squares factoring formula, a^2 - b^2 = (a - b)(a + b):

t^2 - 81 = (t - 9)(t + 9) = 0

Setting each factor equal to zero gives:

t - 9 = 0 --> t = 9

t + 9 = 0 --> t = -9

So, the zeros of the polynomial function P(t) = 3t^3 - 243t are t = 0, t = 9, and t = -9.