Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.
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Transcript of Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.
![Page 1: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/1.jpg)
Warm Up. Find all zeros. Graph. 3 22 7 4x x x
![Page 2: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/2.jpg)
Warm Up. Find all zeros. Graph. 3 22 7 4x x x
4,2,1: FactorsPossible
1 4721 431
0431 141
431 2
xxx
xxx
41 xx
Touches Through
![Page 3: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/3.jpg)
More on Rational Root Theorem
![Page 4: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/4.jpg)
Fundamental Theorem of Algebra
In the complex number system, an nth degree polynomial has precisely “n” zeros.
![Page 5: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/5.jpg)
Find all zeros. 2 6x x
023 xx
23 xx
Goes through both!
![Page 6: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/6.jpg)
Find all zeros. 2 4 4x x
![Page 7: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/7.jpg)
Find all zeros. 2 4 4x x
022 xx
2xMultiplicity of 2.Just Touches.
![Page 8: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/8.jpg)
Find all zeros. 3 4x x
![Page 9: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/9.jpg)
Find all zeros. 3 4x x
042 xx
0x
ix
x
x
2
4
042
2
![Page 10: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/10.jpg)
Find all zeros. 4 16x
![Page 11: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/11.jpg)
Find all zeros. 4 16x
0422
0442
22
xxx
xx
ixixxx 2222
![Page 12: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/12.jpg)
Find all zeros. 3 23 4 12x x x
![Page 13: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/13.jpg)
Find all zeros. 3 23 4 12x x x
34
3432
2
xx
xxx
322 xixix
![Page 14: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/14.jpg)
Find all zeros. 3 24 4 4x x x
![Page 15: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/15.jpg)
Find all zeros. 3 24 4 4x x x
4,2,1: FactorsPossible
Using the remainder theorem wesee that there are NO real solutions!
![Page 16: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/16.jpg)
Find all zeros. 4 36 5 1x x
![Page 17: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/17.jpg)
Find all zeros. 4 36 5 1x x
6
1,3
1,2
1,1
6,3,2,1
1:
FactorsPossible
Synthetic division gets you to the following:
2262
11 2
xxxx
Now use the Quadratic Formulafor this part.
1x
2
1x
0132 2 xx
6
111
6
111
32
13411
ix
x
x
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Find all zeros. 5 3 22 12 8x x x x
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Find all zeros. 5 3 22 12 8x x x x
8,4,2,1: FactorsPossible
Synthetic Division gives
04211 2 xxxx
ixixxx 2221
Multiplicity of 2
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Find all zeros. 4 36 27 20x x x
![Page 21: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/21.jpg)
Find all zeros. 4 36 27 20x x x
20,10,5,4,2,1: FactorsPossible
Synthetic Division gives
5341 2 xxxx
Quadratic Formula completes it.
2
29341
xxx
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Complex Conjugates
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Complex Zeros
Always come in conjugate pairs
i
i
23
23
i
i
i
i
3
3
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Write a polynomial with the given zeros: 2, i, -i
ixixx 2
22
22
12
2
2
23
23
2
22
xxx
xxx
xx
ixx
ixixx
![Page 25: Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.](https://reader035.fdocuments.net/reader035/viewer/2022062409/5697bf7d1a28abf838c84948/html5/thumbnails/25.jpg)
Write a polynomial with the given zeros: 3, 4i
ixixx 443 Remember – they come in pairs!
48163
163
163
443
23
2
22
xxx
xx
ixx
ixixx
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Write a polynomial with the given zeros: 4, 2-3i
ixixx 32324
52298
52164134
1344
963642324
32324
23
223
2
22
xxx
xxxxx
xxx
iixiixxixxx
ixixx
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Find all zeros, given that (2i) is a root.
4 3 2( ) 2 8 12f x x x x x
i2 128121 1286442 iii
1 i22 34 i i6 0i2i2 i4 i6
1 2 3 013
0322
xx
xx
1322 xxixix
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Find all zeros, given that (3 – 5i) is a solution.
3 2( ) 7 40 34f x x x x
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Find all zeros, given that (3 – 5i) is a solution.
3 2( ) 7 40 34f x x x x