Polynomial FunctionsZeros and Graphing

Section 2-2

2

Objectives

• I can find real zeros and use them for graphing

• I can determine the multiplicity of a zero and use it to help graph a polynomial

• I can determine the maximum number of turning points to help graph a polynomial function

3

Complex Numbers

Real Numbers Imaginary Numbers

Rationals Irrational

4

A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa.

A polynomial function of degree n has at most n – 1 turning points and at most n zeros.

Degree (n)

• The degree of a polynomial tells us:

• 1. End behavior – (If n is Odd) Ends in opposite directions– (if n is Even) Ends in same direction

• 2. Maximum number of Real Zeros (n)

• 3. Maximum Number of Turning Points (n-1)

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6

Example: Find all the real zeros and turning points of the graph of f (x) = x

4 – x3 – 2x2.

Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).

The real zeros are x = –1, x = 0, and x = 2.

These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0).

The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.

y

x

f (x) = x4 – x3 – 2x2

Turning pointTurning point

Turning point

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Example: Determine the multiplicity of the zeros of f (x) = (x – 2)3(x +1)4.

Zero Multiplicity Behavior

2

–1

3

4

odd

even

crosses x-axis at (2, 0)

touches x-axis at (–1, 0)

Repeated ZerosIf k is the largest integer for which (x – a)

k is a factor of f (x)and k > 1, then a is a repeated zero of multiplicity k. 1. If k is odd the graph of f (x) crosses the x-axis at (a, 0). 2. If k is even the graph of f (x) touches, but does not cross through, the x-axis at (a, 0).

x

y

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Multiplicity

• Multiplicity is how many times a solution is repeated.

• First find all the factors to a given polynomial. The exponents on each factor determine the multiplicity.

• If multiplicity is ODD, the graph crosses the solution

• If multiplicity is EVEN, the graph just touched or bounces off the solution

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Multiplicity

• (x+2)(x-3)2(x+1)3

• Zeros at (-2,0) (3, 0) and (-1, 0)

• Crosses at (-2, 0)• Touches (3, 0)• Crosses (-1, 0)

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Example: Sketch the graph of f (x) = 4x2 – x4.

1. Write the polynomial function in standard form: f (x) = –x4 + 4x2 The leading coefficient is negative and the degree is even.

2. Find the zeros of the polynomial by factoring.

f (x) = –x4 + 4x2 = –x2(x2 – 4) = – x2(x + 2)(x –2)

Zeros: x = –2, 2 multiplicity 1 x = 0 multiplicity 2

x-intercepts: (–2, 0), (2, 0) crosses through (0, 0) touches only

Example continued

as , )( xfx

x

y

(2, 0)

(0, 0)

(–2, 0)

Putting it all together• 1. Find degree and LC to determine end behavior,

maximum number of real zeros, and maximum number of turning points

• 2. Find y-intercept

• 3. Factor and find all zeros

• 4. Determine multiplicity to determine if graph crosses or touches at the zeros

• 5. Sketch the graph

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12

4 3 2( ) 2f x x x x

: 4

:1

Degree

LCEnd Behavior :

4 possible zeros

3 possible turning points

2Factors are: ( 1)( 2)x x x

Zeros are: (0,0), (1,0), (-2,0)Touch Cross Cross

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Homework

• WS 3-5