Let's analyze each polynomial function to find its zeros.

1. $f(x) = 2x(x - 2)(1 - x)$

• The factors of this function are $2x$, $(x - 2)$, and $(1 - x)$.

• Setting each factor equal to zero:

  1. $2x = 0$ ⟹ $x = 0$
  2. $x - 2 = 0$ ⟹ $x = 2$
  3. $1 - x = 0$ ⟹ $x = 1$

Thus, the zeros are $x = 0$, $x = 1$, and $x = 2$.

2. $f(x) = 2(x + 1)(x - 3)(x - 1)^2$

• The factors are $2(x + 1)$, $(x - 3)$, and $(x - 1)^2$.

• Setting each factor equal to zero:

  1. $x + 1 = 0$ ⟹ $x = -1$
  2. $x - 3 = 0$ ⟹ $x = 3$
  3. $(x - 1)^2 = 0$ ⟹ $x = 1$

Thus, the zeros are $x = -1$, $x = 1$ (with multiplicity 2), and $x = 3$.

3. $f(x) = x^3(x + 2)(x + 3)$

• The factors are $x^3$, $(x + 2)$, and $(x + 3)$.

• Setting each factor equal to zero:

  1. $x^3 = 0$ ⟹ $x = 0$
  2. $x + 2 = 0$ ⟹ $x = -2$
  3. $x + 3 = 0$ ⟹ $x = -3$

Thus, the zeros are $x = 0$, $x = -2$, and $x = -3$.

Final Table of Zeros:

Do you want further details or have any questions?
Here are five related questions:

1. How does the multiplicity of a zero affect the graph of the polynomial?
2. What happens to the end behavior of the polynomial as the degree increases?
3. Can you determine the degree of the polynomial from its factors?
4. How does factoring help in finding the zeros of a polynomial?
5. Why do higher-degree terms dominate the behavior of a polynomial?

Tip: When a zero has even multiplicity, the graph touches the x-axis but does not cross it at that point.