Let's break down the factored form of the polynomial [tex]\( p(x) = x^2(x-2)(x+1)^2 \)[/tex] to determine its zeros and describe the behavior of the graph at each zero.
### Steps:
1. Identify the zeros:
- The zeros of a polynomial are the values of [tex]\( x \)[/tex] where the polynomial equals zero. These are found by setting each factor equal to zero.
2. Determine the zeros and their multiplicities:
- [tex]\( x^2 = 0 \)[/tex] gives us the zero [tex]\( x = 0 \)[/tex] with multiplicity 2 (since the factor is [tex]\( x^2 \)[/tex]).
- [tex]\( (x-2) = 0 \)[/tex] gives us the zero [tex]\( x = 2 \)[/tex] with multiplicity 1.
- [tex]\( (x+1)^2 = 0 \)[/tex] gives us the zero [tex]\( x = -1 \)[/tex] with multiplicity 2.
### Zeros:
- Zero at [tex]\( x = -1 \)[/tex]:
- Since the factor [tex]\( (x+1)^2 \)[/tex] has an exponent of 2 (even), the graph is tangent to the x-axis at [tex]\( x = -1 \)[/tex].
- Zero at [tex]\( x = 0 \)[/tex]:
- Since the factor [tex]\( x^2 \)[/tex] has an exponent of 2 (even), the graph is tangent to the x-axis at [tex]\( x = 0 \)[/tex].
- Zero at [tex]\( x = 2 \)[/tex]:
- Since the factor [tex]\( (x-2) \)[/tex] has an exponent of 1 (odd), the graph crosses the x-axis at [tex]\( x = 2 \)[/tex].
### Listing the Zeros in Order:
- Smallest zero: [tex]\( x = -1 \)[/tex]
- Behavior: Tangent to the x-axis
- Middle zero: [tex]\( x = 0 \)[/tex]
- Behavior: Tangent to the x-axis
- Largest zero: [tex]\( x = 2 \)[/tex]
- Behavior: Crosses the x-axis
### Final List:
- Smallest zero [tex]\( -1 \)[/tex], [tex]\( x \)[/tex]-axis behavior: Tangent
- Middle zero [tex]\( 0 \)[/tex], [tex]\( x \)[/tex]-axis behavior: Tangent
- Largest zero [tex]\( 2 \)[/tex], [tex]\( x \)[/tex]-axis behavior: Crosses
