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Transcript of More on Polynomials - Arizona State University terri/courses/271resources/... x-intercepts, roots,...

  • Outlines

    More on Polynomials

    Terri Miller

    February 16, 2009

    Terri Miller More on Polynomials

  • Outlines Part I: Review of Previous Lecture Part II: Todays Lecture

    Outline of Part I

    1 Summary of the Previous Lecture Vocabulary End Behavior Skills

    2 Extrema local finding local

    Terri Miller More on Polynomials

  • Outlines Part I: Review of Previous Lecture Part II: Todays Lecture

    Outline of Part II

    3 Extrema Global Extrema

    4 Intersections Graphically finding the intersection of two polynomials

    5 Multiplicity multiplicity of a root sketch of the graph graphing with calculator

    6 Long division of Polynomials

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    Part I

    Review of Previous Lecture

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    Vocabulary End Behavior Skills

    Vocabulary

    polynomial

    term

    coefficient

    leading term

    leading coefficient

    degree of a polynomial

    x-intercepts, roots, zeros

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    Vocabulary End Behavior Skills

    End Behavior

    leading term determines

    degree even, both ends go in same direction

    degree odd, the ends go in opposite directions

    leading coefficient > 0, right hand side goes up

    leading coefficent < 0, right hand side goes down

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    Vocabulary End Behavior Skills

    Skills

    Find the x-intercepts and y-intercept of a polynomial function.

    Describe the end behaviors of a polynomial function.

    Determine the minimal degree of a polynomial given its graph.

    Use a graphing utility to find a local maximum or local minimum of a polynomial function.

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    extrema - maxima and minima

    the points B, C , and D are local extrema the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    extrema - maxima and minima

    the points B, C , and D are local extrema

    the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    extrema - maxima and minima

    the points B, C , and D are local extrema the lowest in an area is called a local minimum, C

    the highest in an area is called a local maximum, B and D

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    extrema - maxima and minima

    the points B, C , and D are local extrema

    the lowest in an area is called a local minimum, C

    the highest in an area is called a local maximum, B and D

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    extrema - maxima and minima

    the points B, C , and D are local extrema the lowest in an area is called a local minimum, C the highest in an area is called a local maximum, B and D

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    local extrema - using TI-83/84

    Enter the function in your calculator’s function menu.

    y1 = − 25

    126 (4x4 − 13x3 − 62x2 + 76x − 126)

    graph the function using the window and scale

    x : [−4, 6], xscl = 1; y : [−75, 100], yscl = 25

    use the “calc” key; the second function on the “trace” key

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    local extrema - using TI-83/84

    Enter the function in your calculator’s function menu.

    y1 = − 25

    126 (4x4 − 13x3 − 62x2 + 76x − 126)

    graph the function using the window and scale

    x : [−4, 6], xscl = 1; y : [−75, 100], yscl = 25

    use the “calc” key; the second function on the “trace” key

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    local extrema - using TI-83/84

    Enter the function in your calculator’s function menu.

    y1 = − 25

    126 (4x4 − 13x3 − 62x2 + 76x − 126)

    graph the function using the window and scale

    x : [−4, 6], xscl = 1; y : [−75, 100], yscl = 25

    use the “calc” key; the second function on the “trace” key

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    find the coordinates of C by using the minimum option

    you should get

    find the maximum similarly

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    find the coordinates of C by using the minimum option

    you should get

    find the maximum similarly

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    find the coordinates of C by using the minimum option

    you should get

    find the maximum similarly

    Terri Miller More on Polynomials

  • Previous Lecture Extrema

    local finding local

    find the coordinates of C by using the minimum option

    you should get

    find the maximum similarly

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Part II

    Todays Lecture

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Global Extrema

    3 Extrema Global Extrema

    4 Intersections Graphically finding the intersection of two polynomials

    5 Multiplicity multiplicity of a root sketch of the graph graphing with calculator

    6 Long division of Polynomials

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Global Extrema

    the global maximum is the highest point the graph reaches over its entire domain

    the global minimum is the lowest point the graph reaches over its entire domain

    there may not be a global maximum or minimum

    our example has a global maximum but no global minimum

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Global Extrema

    the global maximum is the highest point the graph reaches over its entire domain

    the global minimum is the lowest point the graph reaches over its entire domain

    there may not be a global maximum or minimum

    our example has a global maximum but no global minimum

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Global Extrema

    the global maximum is the highest point the graph reaches over its entire domain

    the global minimum is the lowest point the graph reaches over its entire domain

    there may not be a global maximum or minimum

    our example has a global maximum but no global minimum

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Global Extrema

    the global maximum is the highest point the graph reaches over its entire domain

    the global minimum is the lowest point the graph reaches over its entire domain

    there may not be a global maximum or minimum

    our example has a global maximum but no global minimum

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Graphically finding the intersection of two polynomials

    3 Extrema Global Extrema

    4 Intersections Graphically finding the intersection of two polynomials

    5 Multiplicity multiplicity of a root sketch of the graph graphing with calculator

    6 Long division of Polynomials

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Graphically finding the intersection of two polynomials

    Example

    Find the intersection of

    y1 = − 25

    126 (4x4 − 13x3 − 62x2 + 76x − 126),

    and y2 = 3x

    2 − 2x2 + x − 12.

    Enter the function(s) in your calculator’s function menu.

    graph the functions using the same window and scale as before

    use the “calc” key; the second function on the “trace” key

    find the coordinates of a point of intersection by using the intersect option

    Terri Miller More on Polynomials

  • Extrema Intersections

    Multiplicity Long division of Polynomials

    Graphically finding the intersection of two polynomials

    Example

    Find the intersection of

    y1 = − 25

    126 (4x4 − 13x3 − 62x2 + 76x − 126),

    and y2 = 3x

    2 − 2x2 + x −