Warm-Up 2/261.

J

10 ∙22∙23=10 ∙ 25=10 ∙32

Rigor:You will learn how to find the real and complex

zeros of polynomial functions.

Relevance:You will be able to use graphs and equations of

polynomial functions to solve real world problems.

2-4 Zeros of Polynomial Functions

Example 1a: List possible rational zeros and determine which, if any are zeros.

𝑓 (𝑥)=𝑥3+2𝑥+1

Step 1 Identify possible rational zeros.

and

Step 2 Test possible rational zeros to determine if they are rational zeros.

𝑝𝑞=± Factors 1

Factors 1=1𝑜𝑟−1

There are no rational zeros.

Example 1b: List possible rational zeros and determine which, if any, are zeros.

𝑔 (𝑥 )=𝑥4+4 𝑥3 −12𝑥− 9Step 1 Identify possible rational zeros.

Step 2 Test possible rational zeros to determine if they are rational zeros.

𝑝𝑞=± Factors 9

Factors 1=±1 , ± 3 ,𝑜𝑟 ± 9

There are two rational zeros at .

0

3

3

1 4 – 12

↓

– 9

– 1 – 3

– 31 – 9

9

0

– 1 – 3

9

0

1 3 – 9

↓ – 3 0

– 31 0

– 3

𝑔 (𝑥)=(𝑥+1) (𝑥+3 )(𝑥2− 3)

𝑥2−3=0𝑥2=3𝑥=±√3  

Example 2: List possible rational zeros and determine which, if any, are zeros.

h (𝑥 )=3 𝑥3− 7 𝑥2 −2 2𝑥+8Step 1 Identify possible rational zeros.

Step 2 Test possible rational zeros to determine if they are rational zeros.

𝑝𝑞=± Factors 8

Factors 3=±1 , ± 2 , ± 4 ± 8 ,± 13 , ± 2

3 , ± 43 ,𝑜𝑟 ± 8

3

There are 3 rational zeros at .

– 22

– 8

– 13

3 – 7 8

↓ – 6 26

43 0

– 2 4

– 1

3 – 13

↓ 12 – 4

03

4

h (𝑥)=(𝑥+2) (𝑥− 4 )(3 𝑥−1)3 𝑥−1=0

3 𝑥=1𝑥=

13

Example 6: Write a polynomial function of least degree with real coefficients in standard form that has the given zeros.

y = (x + 2)(x – 4)[x – (3 – i)][x – (3 + i)]

y = (x + 2)(x – 4)[(x – 3) + i][(x – 3) – i]

y = (x² – 4x + 2x – 8)[(x – 3)² – i(x – 3) + i(x – 3) – i²]

y = (x² – 2x – 8)[(x – 3)² + 1]

y = (x² – 2x – 8)(x² – 6x + 9 + 1)

y = (x² – 2x – 8)(x² – 6x + 10)

y = x4 – 6x3 + 10x² – 2x3 + 12x² – 20x – 8x² + 48x – 80

y = x4 – 8x3 + 14x² + 28x – 80

(x – c)3 + i

Example 7: Write function as (a) the product of linear and irreducible quadratic factors and (b) the product of linear factors. Then (c) list all zeros.𝑘 (𝑥 )=𝑥5 −18 𝑥3+30𝑥2 −19𝑥+30

(a) the product of linear and irreducible quadratic factors

𝑘(𝑥)=(𝑥+5) (𝑥−2 )(𝑥− 3)(𝑥2+1)

𝑥2+1

Example 7: Write function as (a) the product of linear and irreducible quadratic factors and (b) the product of linear factors. Then (c) list all zeros.𝑘 (𝑥 )=𝑥5 −18 𝑥3+30𝑥2 −19𝑥+30

(b) the product of linear factors

𝑘(𝑥)=(𝑥+5) (𝑥−2 )(𝑥− 3)(𝑥2+1)

𝑥2+1=0𝑥2=−1𝑥=±√−1𝑥=± 𝑖

𝑘(𝑥)=(𝑥+5) (𝑥−2 )(𝑥− 3)(𝑥+ 𝑖)(𝑥−𝑖)(c) List all zerosThere are 5 zeros: .

Example 8: Use given zeros to find all complex zeros. Then write the linear factorization of the function.𝑝 (𝑥 )=𝑥4 − 6𝑥3+20 𝑥2− 22𝑥− 13 given 2 −3 𝑖as a zero of 𝑝 .

20

24 + 3i

– 4 – 3i

1 – 6 – 22

↓

– 13

2 – 3i – 17 + 6i

3 + 6i1 2 + 3i

13

0

2 – 3i

3 + 6i

– 2 – 3i

– 2

1 – 4 – 3i 2 + 3i

↓ 2 + 3i – 4 – 6i

– 11 0

2 + 3i

𝑝 (𝑥 )= [𝑥− (2 −3 𝑖 ) ] [𝑥− (2+3 𝑖 ) ] (𝑥2 −2 𝑥−1)

𝑥2−2 𝑥−1

Example 8: Use given zeros to find all complex zeros. Then write the linear factorization of the function.𝑝 (𝑥 )=𝑥4 − 6𝑥3+20 𝑥2− 22𝑥− 13 given 2 −3 𝑖as a zero of 𝑝 .

𝑝 (𝑥 )= [𝑥− (2 −3 𝑖 ) ] [𝑥− (2+3 𝑖 )] (𝑥2 −2 𝑥−1)

𝑥=−𝑏±√𝑏2 − 4𝑎𝑐2𝑎

𝑥=2 ±√4 − 4 (1)(−1)

2𝑥=2 ±√8

2

𝑥=2 ±2√22

𝑥=1 ±√2

𝑝 (𝑥 )= [𝑥− (2 −3 𝑖 ) ] [𝑥− (2+3 𝑖 ) ] [𝑥− ( 1+√2 ) ] [𝑥− ( 1−√2 ) ]

√−1math!

2-4 Assignment: TX p127, 4-16 EOE & 32-52 EOE