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To analyze : The polynomial function
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Answer to Problem 11RE
The polynomial function
The y -intercept is 3.
The x -intercept is zero 1, -1 and -3.
Explanation of Solution
Given information:
Formula used:
The graph of a polynomial function
Degree of a polynomial function
Maximum number of turning points :
At a zero even multiplicity: The graph of f touches the x -axis.
At a zero odd multiplicity : The graph of f crosses the x -axis.
For large
Calculation:
Consider ,
Step 1.
Re-writing the polynomial is,
The polynomial function f is of degree 4. The graph of f behaves like
Step 2.
The y −intercept :-
The x -intercept :-
Step3.
The zeros of f are 1, -1 and -3. The zero 1 is a zero of multiplicity 3, so the graph of f crosses the x - axis at
Step4.
Since the degree of polynomial function is 4. Therefore, the graph of the function will have at most
Step5.
The x -intercept are 1, -1 and -3.
The behavior of the graph of f near each x -intercept are as follows:-
Near
A parabola that opens up .
Near
The line with slope 8.
Near
The line with slope -32.
Step6.
Using Step1- Step5 graph is drawn showing the x - intercept and y -intercept. With the help of graph the behavior of each x −intercept can be observed.
Figure1.
Hence , the polynomial function f has been analyzed.
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