Matt Wolf



Properties of Polynomial Functions

degree – leading coefficient –

Zeros of Polynomial Functions

If[pic] is a zero of the polynomial function[pic], then the following statements are true:

• (c, 0) is an x-intercept of the graph of the function.

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.

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.

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[pic] [pic] [pic]

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.

|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 to left and rises to right |Graph rises to left and falls to right |

|[pic] |[pic] |[pic] |[pic] |

|Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |Trick: Think of [pic] |

|[pic] |[pic] |[pic] |[pic] |

Examples: Apply the Leading Coefficient Test to describe the end behavior of a graph of the functions.

[pic] [pic]

[pic]

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