Warm-Up 2/26
description
Transcript of Warm-Up 2/26
Warm-Up 2/261.
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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