Concept explainers
In Problems
List each real zero and its multiplicity.
Determine whether the graph crosses or touches the x-axis at each x-intercept.
Determine the maximum number of turning points on the graph.
Determine the end behaviour; that is, find the power function that the graph of
(a)
Real zeros and its multiplicity for the polynomial function
Answer to Problem 51AYU
Solution:
The polynomial function
Explanation of Solution
Given Information:
The polynomial function
If
That is,
Using the zero product rule,
To find zeros
To get rid of square, take the square root on both sides,
Subtract 4 from both sides,
There is no real number, where the square is negative
Therefore,
Now, find
To get rid of cube, take the cube root on both sides,
Add 5 on both sides,
Hence, the function has real zero 5.
The multiplicity of real zero is the number of times the corresponding factor occurs.
In the given polynomial
Therefore, the polynomial function
(b)
Whether the graph of polynomial function
Answer to Problem 51AYU
Solution:
The polynomial function
Explanation of Solution
Given Information:
The polynomial function
The x-intercepts satisfy the equation
That is,
From part (a), the zero of the polynomial function that is x-intercepts is only 5. The real zero 5 has odd multiplicity 3
If the polynomial has real zero
Therefore, the polynomial function
(c)
The maximum number of turning points on the graph of polynomial function
Answer to Problem 51AYU
Solution:
The maximum number of turning points on the graph of polynomial function
Explanation of Solution
Given Information:
The polynomial function
The polynomial function
To simplify this polynomial, first distribute
By using the distributive property,
Add like terms,
Now, distribute
By using the distributive property,
Add the like terms,
Now, distribute
By using the distributive property,
Add the like terms,
Now, distribute
By using the distributive property,
Add the like terms,
Now, distribute 7 inside the bracket,
This polynomial has degree 7.
If
Hence, the maximum number of turning points with degree 7 is
Therefore, the maximum number of turning points on the graph of polynomial function
(d)
The end behavior, that is the power function for that graph of
Answer to Problem 51AYU
Solution:
The end behavior that is the power function for that graph of
Explanation of Solution
Given Information:
The polynomial function
The end behavior of the graph of polynomial function
From part (c), the polynomial function
Hence, the power function is
Here, the polynomial has positive leading coefficient 7 and odd degree that is 7. Hence, as
Therefore, the end behavior, that is the power function for that graph of
Chapter 4 Solutions
Precalculus
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