Section 4.1: Polynomial Functions and...

MA 180 Precalculus (Scott) Chapter 4: Polynomial & Rational Functions

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Section 4.1: Polynomial Functions and Models

Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial Function and Their Multiplicity 4. Analyze the Graph of a Polynomial Function 5. Build Cubic Models from Data


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Function Factors Graph Zeros Multiplicity

( ) 2 3 10f x x x= − −

( ) 2 4 4f x x x= + +

( ) 3f x x x= −

( ) 3 26 9f x x x x= − +

( ) 22 1f x x= +


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( ) nf x ax= , n is even ( ) nf x ax= , n is odd

( ) 212f x x= ( ) 5f x x=

( ) 4f x x= ( ) 3f x x=

( ) 23f x x= − ( ) 512f x x= −

( ) 612f x x= − ( ) 32f x x= −

1. 2. 3. 4. 1. 2. 3. 4.


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Find a polynomial f of degree 3 whose zeros are -4, -2, and 3.


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Determine the polynomial function whose graph is given.


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Summary Analyzing the Graph of a Polynomial Function Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8:


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(0,9)

1. Determine [algebraically] the quadratic

function whose graph is given. Write in standard form: 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 +𝑐,𝑎 ≠ 0You must show work to receive credit.

2. Both graphs below are of the same polynomial function [in different viewing

windows]. The one on the left is not a complete graph.

Write an equation of the graph pictured that includes the points (0, 0), (-5, 0),

(4, 0), (-1, 0) and (1, 216). Leave your equation in factored form.

-200

-100

0

100

200

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-500

500

1500

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


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3. The following statements about the polynomial function f are true. (5 points)

• f has exactly three zeros. • The graph of f has the following end behavior: as x → −∞, f(x) → − ∞ and as ∞→x ,

∞→)(xf .

For each of the following statements circle T if the statement must always be true, F if the is always false, or S if the statement is sometimes false and sometimes true.

T F S The leading coefficient is negative. T F S f(x) is an odd function. T F S The graph of f has at most three local extrema (local maxima/minima). T F S f(x) is a third degree polynomial function. T F S f(x) has at most two x-intercepts.

4.


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Finding Zeros of Polynomial Functions (4.2 & 4.3) EXAMPLE: Using the Remainder Theorem Is (𝑥 − 1) a factor of 𝑓(𝑥) = 2𝑥3 + 11𝑥2 − 7𝑥 − 6? Is (𝑥 + 4) a factor?


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Find the real zeros of the polynomial function 𝑓(𝑥) and completely factor. 1. 𝑓(𝑥) = 9𝑥3 + 7𝑥2 − 45𝑥 − 35 2. 𝑓(𝑥) = 7𝑥4 + 41𝑥3 + 78𝑥2 + 44𝑥 − 8 3. 𝑓(𝑥) = 7𝑥3 + 13𝑥2 − 9𝑥 + 1


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Find the real zeros of f rounded to decimal places:


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Section 4.3 Complex Zeros; Fundamental Theorem of Algebra Find the zeros of the polynomial 𝑓(𝑥) = 𝑥2 + 1 and factor.


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Find a Polynomial Function with Specified Zeros Find a polynomial function 𝑓 of degree 4 whose coefficients are real numbers and that has zeros specified zeros: 1, 1, and −4 + 𝑖 Hint: If 𝑟 = 𝑎 + 𝑏𝑖 and �̅� = 𝑎 − 𝑏𝑖, then when the linear factor (𝑥 − 𝑟) and (𝑥 − �̅�) are multiplied, we have

(𝑥 − 𝑟)(𝑥 − �̅�) = 𝑥2 − (𝑟 + �̅�)𝑥 + 𝑟�̅�


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Find the complex zeros of 𝑓(𝑥) = 7𝑥3 − 16𝑥2 + 74𝑥 − 20 and completely factor.


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Find the remaining zeros of ℎ(𝑥) = 𝑥4 − 4𝑥3 + 32𝑥 − 13; zero: 3 − 2𝑖 Use synthetic or long division.

Fundamental Theorem of Algebra

Every complex polynomial function f (x) of degree n > 1 has at least one complex zero.

Theorem

Every complex polynomial function f (x) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form

Conjugate Pairs Theorem

Let f (x) be a complex polynomial whose coefficients are real numbers. If r = a + biis a zero of f, then the complex conjugate

Corollary

A complex polynomial f of odd degree with real coefficients has at least one real zero.

is also a zero of f.

Section 4.6

Section 4.1: Polynomial Functions and...

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Transcript of Section 4.1: Polynomial Functions and...