Question

find the zeros for the given polynomial function and give the multiplicity for each zero. state whether the graph crosses the x - axis or touches the x - axis and turns around at each zero.
f(x)=x^3 - 4x^2 + 4x

determine the zero(s), if they exist.
the zero(s) is/are 0,2.
(type integers or decimals. use a comma to separate answers as needed.)

determine the multiplicities of the zero(s), if they exist. select the correct choice below and, if necessary, fill in the answer box(es) within your choice.

a. there are three zeros. the multiplicity of the smallest zero is. the multiplicity of the largest zero is. the multiplicity of the other zero is. (simplify your answers.)
b. there is one zero. the multiplicity of the zero is. (simplify your answer.)
c. there are two zeros. the multiplicity of the smallest zero is. the multiplicity of the largest zero is. (simplify your answers.)

Explanation:

Step1: Factor the polynomial

First, factor out the common factor $x$ from $f(x)=x^{3}-4x^{2}+4x$, we get $f(x)=x(x^{2}-4x + 4)$. Then factor the quadratic $x^{2}-4x + 4=(x - 2)^{2}$. So $f(x)=x(x - 2)^{2}$.

Step2: Find the zeros and multiplicities

Set $f(x)=0$. Then $x(x - 2)^{2}=0$. By the zero - product property, $x=0$ or $(x - 2)^{2}=0$. The zero $x = 0$ has multiplicity 1 (since the factor $x$ has an exponent of 1), and the zero $x = 2$ has multiplicity 2 (since the factor $(x - 2)$ has an exponent of 2).

Answer:

C. There are two zeros. The multiplicity of the smallest zero is 1. The multiplicity of the largest zero is 2.