Matt Wolf - Central Bucks School District



|Learning Goals: |At the completion of the lesson, you will be able to… |

| |Identify the degree and leading coefficient of polynomial functions |

| |Calculate the zeros of polynomial functions algebraically |

| |Define and apply multiplicity to zeros of polynomial functions |

| |Apply Leading Coefficient Test to describe end behavior of graphs |

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.

-----------------------

[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]

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