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Cassandra was given a polynomial function and asked to describe its zeros. The first thing Cassandra did was factor the polynomial expression of this function. The factored form is shown:

[tex]\[ p(x) = x^2(x-2)(x+1)^2 \][/tex]

Based on Cassandra's factorization of [tex]\( p(x) \)[/tex], list the zeros in order from least [tex]\( x \)[/tex]-value to greatest [tex]\( x \)[/tex]-value.

After each zero, state whether the graph crosses the [tex]\( x \)[/tex]-axis at that point or is tangent to the [tex]\( x \)[/tex]-axis at that point.

- Smallest zero: [tex]\(\square\)[/tex]
- [Select] Crosses the [tex]\( x \)[/tex]-axis
- [Select] Is tangent to the [tex]\( x \)[/tex]-axis

- Middle zero: [tex]\(\square\)[/tex]
- [Select] Crosses the [tex]\( x \)[/tex]-axis
- [Select] Is tangent to the [tex]\( x \)[/tex]-axis

- Greatest zero: [tex]\(\square\)[/tex]
- [Select] Crosses the [tex]\( x \)[/tex]-axis
- [Select] Is tangent to the [tex]\( x \)[/tex]-axis

Sagot :

GinnyAnsweridn
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
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