5-2 Polynomials, Linear Factors, & Zeros

Today’s Objective:

I can write and graph a polynomial function

𝑦=(𝑥−)(𝑥−)(𝑥−)

Roots, Zeros & x-intercepts𝑷 (𝒙 )=𝒂𝒏 𝒙

𝒏+𝒂𝒏−𝟏 𝒙𝒏−𝟏+⋯+𝒂𝟏𝒙+𝒂𝟎

Factor Theorem is a linear factor of the polynomial

if and only ifb is a zero of the polynomial function

Find the zeros

𝑦=(𝑥−1 ) (𝑥+2 )(𝑥−3)Write the polynomialgiven the zeros:

𝑦=(𝑥2−4)(𝑥−3)𝑦=𝑥3−3𝑥2−4 𝑥+12

2+2 31 ,−2 ,3

𝑦=𝑥 (𝑥+5 ) (𝑥−7 )0 ,−5 ,7

𝑦=𝑥 (𝑥−4 )(𝑥+6)0 ,4 ,−6

𝑦=𝑥 (𝑥2+2 𝑥−24)𝑦=𝑥3+2𝑥2−24 𝑥

-5 -4 -3 -2 -1 1 2 3 4 5-25

-20

-15

-10

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25

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x

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x

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Graphing with zeros

1. Find and plot the zeros2. Sketch end behavior3. Pick easy midpoints

between zeros to estimate turning point

𝑓 (𝑥)=𝑥(𝑥−4)(𝑥+3)

Zeros: 0 ,4 ,−3𝑓 (−2)=¿−2 (−2−4)(−2+3)

¿12𝑓 (2)=¿2(2−4)(2+3)

¿−20

−2 ,−2 ,2 ,3

Zeros with Multiplicity

1. Find and plot the zeros2. Sketch end behavior3. Pick easy midpoints

between zeros to estimate turning point

𝑓 (𝑥)=(𝑥+2)(𝑥+2)(𝑥−2)(𝑥−3)

Zeros:

𝑓 (0)=¿¿24

𝑓 (2.5)=¿¿−5

𝑓 (𝑥)=(𝑥+2)2(𝑥−2)(𝑥−3)

(0+2)2(0−2)(0−3)

(2.5+2 )2(2.5−2)(2.5−3)-5 -4 -3 -2 -1 1 2 3 4 5

-10

-5

5

10

15

20

25

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y

-5 -4 -3 -2 -1 1 2 3 4 5-10

-5

5

10

15

20

25

x

y

-5 -4 -3 -2 -1 1 2 3 4 5-10

-5

5

10

15

20

25

x

y

-5 -4 -3 -2 -1 1 2 3 4 5-10

-5

5

10

15

20

25

x

y

• Even multiplicity turns graph at zero

• Odd multiplicity pauses graph zero

W.S. 5-1/5-2 Polynomials, Linear Factors & Zeros