Zeros of Polynomials

1
Chapter 6

• Section 2
• Zeros of Polynomials

2
Definition

• Consider
• A zero of a polynomial is a value of x that
  causes p(x) to be zero.

3
Quick Reminder

• Zero Product PropertyIf a and b are any real
  numbers such that ab 0then either a 0 or
  b 0

4
Difference of Two Squares
5
More Examples
6
x-intercepts and zeros
7

• i. Copy the image onto your graph paper. Label
  and scale your axes, then label each x-intercept
  with its coordinates.
• ii. Identify the zeros of the polynomial.
• iii. Identify the linear factors of the
  polynomial.

8

• i. Copy the image onto your graph paper. Label
  and scale your axes, then label each x-intercept
  with its coordinates.
• ii. Identify the zeros of the polynomial.
• iii. Identify the linear factors of the
  polynomial.

9

• i. Factor the polynomial to obtain the zeros.
  Show your work.
• ii. Set up a coordinate system on graph paper.
  Label and scale the horizontal axis. Use the
  zeros and end-behavior to help sketch the graph
  of the polynomial without the use of a
  calculator.
• iii. Verify your result with a graphing
  calculator.

10
Chapter 6

• Section 3
• Extrema and Models

11
Extreme Values

• Where a polynomial function makes a turn there is
  a relative maximum or a relative minimum. These
  values are the extrema of the function.

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• i. Without the aid of a calculator, use an
  algebraic technique to identify the zeros of the
  given polynomial. Factor if necessary.
• ii. On graph paper, set up a coordinate system.
  Label each axis, but scale only the x-axis. Use
  the zeros and the end-behavior to draw a rough
  graph of the given polynomial without the aid of
  a calculator.
• iii. Classify each local extrema as a relative
  minimum or relative maximum. At this point it is
  not necessary to find the coordinates of the
  relative extrema.
• Using the MAX or MIN capabilities of your
  calculator give estimates of the values of the
  ordered pairs that make up the extrema.

13
Make a Box

• A rectangular piece of cardboard measures 8
  inches by 12 inches. Schuyler cuts four smaller
  squares from each corner of the cardboard,
  tossing the material aside. He then bends up the
  sides of the remaining cardboard to form an open
  box with no top. Find the dimensions of the
  squares cut from each corner of the original
  piece of cardboard so that Schuyler maximizes the
  volume of the resulting box.

14

• Restrict the graph of the parabola y 4 - x2/4
  to the first quadrant, then inscribe a triangle
  inside the parabola, as shown in the figure that
  follows.
• What length of base and height of the triangle
  maximize the area?