Matt Wolf
The Rational Root Test
If the polynomial function[pic]has integer coefficients, then every rational (non-complex) zero of the function has the form [pic] where p is a factor of the constant term and q is a factor of the leading coefficient.
Example: Find the rational zeros of [pic].
1) Try to factor the polynomial.
2) Identify the constant term ( ___ ) and the leading coefficient ( ___ ).
3) Create a fraction of the form: factors of the constant term
factors of the leading coefficient
4) From this fraction, create a list of all possible rational zeros.
5) Use synthetic division to divide the polynomial by each of these possible rational zeros. If the quotient has a remainder, the value is not a zero. If the quotient does not have a remainder, the value is a zero.
Example: Find the rational zeros of [pic].
Multiplicity
If [pic] is a polynomial function with factor[pic], then a is called a repeated zero if the factor [pic] occurs more than once in the linear factorization of[pic]. The number of times the factor [pic] occurs is called the multiplicity of [pic].
• If the multiplicity of [pic]is odd, the graph of [pic]crosses the x-axis at[pic].
• If the multiplicity of [pic]is even, the graph of [pic]touches (but does not cross) the x-axis at[pic].
Examples: Find all zeros of the following functions. Determine the multiplicity of each zero to determine whether the graph of the function crosses or touches the x-axis at each.
The Leading Coefficient Test
The end behavior of a graph of a polynomial function can be determined using the Leading Coefficient Test which identifies four basic cases.
Example: Sketch the graph of the function [pic].
1) Apply the Leading Coefficient Test to determine the end behavior of the graph.
[pic] as [pic] [pic] as [pic]
2) Find the zeros of the function. Identify the multiplicity of each zero to determine if the graph crosses or touches the x-axis at this point.
|Zeros | | | | |
|Multiplicity | | | | |
|Graph crosses | | | | |
|or touches x-axis | | | | |
3) Arrange the zeros in order to form test intervals. Choose a test value in each test interval. Evaluate the *factored form* of the function at each test value to determine if the function is positive or negative in the interval.
|Test Interval | | | | | |
|Test Value | | | | | |
|Sign of [pic] | | | | | |
|(at Test Value) | | | | | |
|Graph is Above | | | | | |
|or Below x-axis | | | | | |
4) Sketch the graph with smooth, continuous curves using the information obtained in Steps 1-3.
-----------------------
|degree is even |degree is even |degree is odd |degree is odd |
|leading coefficient is pos. |leading coefficient is neg. |leading coefficient is pos. |leading coefficient is neg. |
|Graph rises to left and right |Graph falls to left and right |Graph falls left, rises right |Graph rises left, falls right |
|[pic] |[pic] |[pic] |[pic] |
|Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |
Examples: Apply the Leading Coefficient Test to describe the end behavior of a graph of the functions.
[pic] [pic]
[pic]
[pic] [pic] [pic]
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