Using a graphing utility, graph and approximate the zeros and their multiplicity.
A.x=0,multiplicity 2;x=7,multiplicity 3
B. x=0,multiplicity 2; x=7,multiplicity 1; x=-7, multiplicity 2
C. x=0,multiplicity 3;x=0,multiplicity 2
D.x=0,multiplicity 3;x=-7, multiplicity 2
E.x=0,multiplicity 2;x=7, multiplicity 2; x=-7, multiplicity 1
To graph and approximate the zeros and their multiplicities using a graphing utility, you need to follow the steps below:
1. Open your graphing utility software, such as Desmos or Graphing Calculator.
2. Enter the equation or function representing the polynomial. In this case, there is no equation or function given, but we are provided with the zeros and their multiplicities.
3. For each zero, plot a point on the x-axis corresponding to its value. The multiplicities will determine the behavior of the graph near those zeros.
4. To describe the multiplicity, consider the powers of factors that create the zeros. For example, if a zero has a multiplicity of 2, it means that the corresponding factor is squared.
5. Based on the multiplicities, you can observe the graph's behavior near each zero. For example, if a zero has an odd multiplicity, the graph will cross the x-axis at that zero. If the multiplicity is even, the graph will touch or bounce off the x-axis at that zero.
6. Finally, connect the points on the graph to get an approximation of the shape of the polynomial.
Analyzing the given answer choices:
A. x=0, multiplicity 2; x=7, multiplicity 3
This means the graph will touch or bounce off the x-axis at x=0 and cross through the x-axis at x=7. The provided multiplicities are consistent with this behavior, so this choice could be correct.
B. x=0, multiplicity 2; x=7, multiplicity 1; x=-7, multiplicity 2
This choice includes additional zero at x=-7 with multiplicity 2. However, this zero is not given in the question, so it does not match with the problem statement. Therefore, this choice is not correct.
C. x=0, multiplicity 3; x=0, multiplicity 2
This choice has two zeros at x=0, but it does not have the zero with multiplicity 3 at x=7. So, this choice does not match with the given information and is not correct.
D. x=0, multiplicity 3; x=-7, multiplicity 2
This choice includes a zero at x=-7, but it is not given in the problem statement. Therefore, this choice does not match with the information provided and is not correct.
E. x=0, multiplicity 2; x=7, multiplicity 2; x=-7, multiplicity 1
This choice includes all the zeros and their multiplicities mentioned in the problem statement, as well as an additional zero at x=-7 with multiplicity 1. However, this additional zero is not given in the question, so this choice does not match with the information provided and is not correct.
Based on the analysis, the answer choice that matches the given information is:
A. x=0, multiplicity 2; x=7, multiplicity 3