To determine the zeros of the polynomial function [tex]\( f(x) = x^3 - 9x^2 + 20x \)[/tex], we need to find the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
We start with the given polynomial:
[tex]\[ f(x) = x^3 - 9x^2 + 20x. \][/tex]
To find the zeros, we set the polynomial equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^3 - 9x^2 + 20x = 0. \][/tex]
First, we can factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^2 - 9x + 20) = 0. \][/tex]
This gives us one zero immediately:
[tex]\[ x = 0. \][/tex]
Next, we need to solve the quadratic equation [tex]\( x^2 - 9x + 20 = 0 \)[/tex] for the remaining zeros. We can factor the quadratic expression:
[tex]\[ x^2 - 9x + 20 = (x - 4)(x - 5). \][/tex]
Setting each factor equal to zero, we solve for [tex]\( x \)[/tex]:
[tex]\[ x - 4 = 0 \quad \Rightarrow \quad x = 4, \][/tex]
[tex]\[ x - 5 = 0 \quad \Rightarrow \quad x = 5. \][/tex]
Thus, the zeros of the polynomial function [tex]\( f(x) = x^3 - 9x^2 + 20x \)[/tex] are [tex]\( x = 0 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( x = 5 \)[/tex].
So, the correct answer is:
[tex]\[ 0, 5, 4. \][/tex]
