Solve by Factoring: 4 2( ) 4 11 6f x x x

234 22 xx

2

3

4

3

34

2

2

x

x

x

2

22

x

x

Solve by completing the Square:

2( ) 3 6 1f x x x

3

321

3

321

3

41

3

41

134

12331

231

631

2

2

2

2

2

2

x

x

x

x

x

xx

xx

xx

Rational Root Test

Remainder Theorem

Remainder = f(k) Example:

f(2)=

3 23 2 5 5 2x x x x

31

510824

5104283

5252223 23

2686

55232

311343

Find the remainder:

3 28 2 5 4 2x x x x

86

410864

4104288

42522282 23

f

Therefore, x = -2 is NOT a root! FactorTheorem

Factor Theorem

f(x) has a factor (x-k) iff f(k)=0.

Rational Zero Test

If f(x)=anxn + an-1xn-1 +… + a1x + a0

Then the possible rational roots are

Factors of the last term (a0) over the factors of the first term (an)

termfirstoffactors

termlastoffactors

Example 523 23 xxxf

3,1 5,1

3

5,3

1,5,1: Possible

Find all real roots:3 2( ) 1f x x x x

11

1:

Possible

1 1111

0101

101

11

111

11 2

xx

xxx

xx

Mult. of 2Touches.

Goes Through

x y

0 1

Find all real roots:3 2( ) 2 3 8 3f x x x x

Find all real roots:

1 3832

0352

352

3121

3521 2

xxx

xxx

Goes Through ALL

x y

0 3

3 2( ) 2 3 8 3f x x x x

2

3,2

1,3,1

2,1

3,1:

Possible

3,2

11 xxx

Find all real roots:

1 65551

061161

61161

x y

0 -6

4 3 2( ) 5 5 5 6f x x x x x

6,3,2,11

6,3,2,1:

Possible

1 651

3211

6511

611612

23

xxxx

xxxx

xxxx

651

3211 xxxx

All Go Through

Find all real roots:4 3 2( ) 3 6f x x x x x

Find all real roots:Do NOT Graph.

1 63111

06321

6321

6,3,2,11

6,3,2,1:

Possible

2 602

321

63212

23

xxx

xxxx

301

4 3 2( ) 3 6f x x x x x

321 2 xxx

NOT Real!

Find all real roots:Do NOT Graph.

5 4 3 2( ) 3 5 15 4 12f x x x x x x

12,6,4,3,2,11

12,6,4,3,2,1:

Possible

1

01216141

1216141

12415531

1012431

12431

2

0651

12102

32211

65211

124311

121641

2

23

234

xxxxx

xxxxx

xxxxx

xxxxx

32211 xxxxx

Find all real roots: 3 2( ) 2 3 11 6f x x x x

Find all real roots: 3 2( ) 2 3 11 6f x x x x

Complex Numbers

Imaginary Unit (i) = 1

Complex Numbers:

Consists of a real number plus an imaginary number

Looks like: a + bi

Can also be called an imaginary number

If a = 0, then it’s a pure imaginary number

Simplify:

7 4

7

71

i

i2

41

Simplify:

3 6 6 8

23

291

18

632

i

ii

34

3161

48

862

i

ii

Simplify:

26i i 5 2 3 8i i

i

i

i

61

16

16

i108

Simplify:

2 3 5i i 7 3 2 9i i

i

i

i

ii

610

106

1106

106 2

i65

Simplify:

3 4 2 7i i 3 2 3 2i i

i

i

iii

1334

128136

288216 2

13

49

149

49 2

i

Simplify:

3 2 4i i 23 2i

i

i

iii

514

12512

28312 2

i

i

ii

ii

125

14129

4129

23232

Simplify:

3 2

4

i

i

6

1 2

i

i

17

1110

116

121112

16

28312

4

4

4

23

2

2

i

i

i

iii

i

i

i

i

5

134

141

1213641

2126

21

21

21

6

2

2

i

ii

iii

i

i

i

i