Answer :
To determine the behavior of the graph of the polynomial [tex]\( y = -x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 \)[/tex] at each of its zeros, we look at the roots and their multiplicities. Here's how to analyze this:
1. Identify the Zeros of the Polynomial:
The roots or zeros of a polynomial are the values of [tex]\(x\)[/tex] for which the polynomial equals zero. For this polynomial, the zeros are [tex]\( -3, -3, -3, -1, 2, \)[/tex] and [tex]\( 2 \)[/tex].
2. Determine the Multiplicity of Each Zero:
- The zero [tex]\(-3\)[/tex] has a multiplicity of 3.
- The zero [tex]\(-1\)[/tex] has a multiplicity of 1.
- The zero [tex]\(2\)[/tex] has a multiplicity of 2.
3. Analyze the Behavior at Each Zero:
- Zero [tex]\(-3\)[/tex] with a multiplicity of 3:
A zero with multiplicity 3 resembles the behavior of a cubic function at that zero. The graph will "flatten" and cross the x-axis at this point.
- Zero [tex]\(-1\)[/tex] with a multiplicity of 1:
A zero with multiplicity 1 resembles the behavior of a linear function. The graph will cross the x-axis at this point and won't flatten.
- Zero [tex]\(2\)[/tex] with a multiplicity of 2:
A zero with multiplicity 2 resembles the behavior of a quadratic function. The graph will touch the x-axis at this point and turn back, creating what’s typically called "a bounce."
Given this analysis, the answer is:
C. one resembles a cubic function, one resembles a linear function, and one resembles a quadratic function.
1. Identify the Zeros of the Polynomial:
The roots or zeros of a polynomial are the values of [tex]\(x\)[/tex] for which the polynomial equals zero. For this polynomial, the zeros are [tex]\( -3, -3, -3, -1, 2, \)[/tex] and [tex]\( 2 \)[/tex].
2. Determine the Multiplicity of Each Zero:
- The zero [tex]\(-3\)[/tex] has a multiplicity of 3.
- The zero [tex]\(-1\)[/tex] has a multiplicity of 1.
- The zero [tex]\(2\)[/tex] has a multiplicity of 2.
3. Analyze the Behavior at Each Zero:
- Zero [tex]\(-3\)[/tex] with a multiplicity of 3:
A zero with multiplicity 3 resembles the behavior of a cubic function at that zero. The graph will "flatten" and cross the x-axis at this point.
- Zero [tex]\(-1\)[/tex] with a multiplicity of 1:
A zero with multiplicity 1 resembles the behavior of a linear function. The graph will cross the x-axis at this point and won't flatten.
- Zero [tex]\(2\)[/tex] with a multiplicity of 2:
A zero with multiplicity 2 resembles the behavior of a quadratic function. The graph will touch the x-axis at this point and turn back, creating what’s typically called "a bounce."
Given this analysis, the answer is:
C. one resembles a cubic function, one resembles a linear function, and one resembles a quadratic function.
