Warm Up. Find all zeros. Graph. 3 22 7 4x x x

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

More on Rational Root Theorem

Fundamental Theorem of Algebra

In the complex number system, an nth degree polynomial has precisely “n” zeros.

Find all zeros. 2 6x x

023 xx

23 xx

Goes through both!

Find all zeros. 2 4 4x x

Find all zeros. 2 4 4x x

022 xx

2xMultiplicity of 2.Just Touches.

Find all zeros. 3 4x x

Find all zeros. 3 4x x

042 xx

0x

ix

x

x

2

4

042

2

Find all zeros. 4 16x

Find all zeros. 4 16x

0422

0442

22

xxx

xx

ixixxx 2222

Find all zeros. 3 23 4 12x x x

Find all zeros. 3 23 4 12x x x

34

3432

2

xx

xxx

322 xixix

Find all zeros. 3 24 4 4x x x

Find all zeros. 3 24 4 4x x x

4,2,1: FactorsPossible

Using the remainder theorem wesee that there are NO real solutions!

Find all zeros. 4 36 5 1x x

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

Find all zeros. 5 3 22 12 8x x x x

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

Find all zeros. 4 36 27 20x x x

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

Complex Conjugates

Complex Zeros

Always come in conjugate pairs

i

i

23

23

i

i

i

i

3

3

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

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

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

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

Find all zeros, given that (3 – 5i) is a solution.

3 2( ) 7 40 34f x x x x

Find all zeros, given that (3 – 5i) is a solution.

3 2( ) 7 40 34f x x x x