Ch 2.5: The Fundamental Theorem of AlgebraTheorem: In the complex number system, every nth

degree polynomial has n zerosComplex system: Real AND Imaginary numbers

Finding all zeros 1. Use Descartes to determine possible solutions2. Use the calculator to find all rational zeros3. Use synthetic division to get the problem to a

quadratic equation4. Use the quadratic formula to find the remaining zeros

***If a solution is imaginary, then its conjugate is a solution as well***

Ex:

1. Find possible number of positive and negative zeros

2. Graph3. Find the zeros**x=1 has multiplicity of 2 to

give 2 positive zeros**

Continued….

5 3 2( ) 2 12 8f x x x x x

X = -2 X = 1 X = 1

Positive: 2 or 0

Negative: 1

Imaginary: 4 or 2

Total Solutions: 5

5. Synthetic divide one zero at a time

6. Use the quadratic formula to find the final zeros

5 3 2( ) 2 12 8f x x x x x 2 | 1 0 1 2 12 8

-2 4 -10 16 -8

1 -2 5 -8 4 0

1 | 1 -2 5 -8 4

1 -1 4 -4

1 -1 4 -4 0

1 | 1 -1 4 -4

1 0 4

1 0 4 0

2 0 4x x 0 0 4(1)(4)

2(1)x

16

2

4

2

ix

1, 1, 2, 2 , 2x i i ALL SOLUTIONS

2i

Sometimes you must factor…

1. Graph give 2 irrational zeros, so try factoring!

2. Set each factor equal to zero

4 2 20x x 2 2( 5)( 4)x x

2 5 0x 2 4 0x 2 5x

5x 2 4x

4x 2x i

ALL SOLUTIONS

5, 5, 2 , 2x i i

Find all roots when given one complex root

1. Change it to a factor

2. Multiply it by its conjugate (because it is a root as well)

3. Use Long division to get the remaining quadratic equation

4. Factor the equation to find the zeros or use the quadratic formula

Ex: Find all the roots of if 1 + 3i is a zero

1. Turn it into a factor

2. Multiply by its conjugate

3. Divide by the new factor

Continued…

4 3 2( ) 3 6 2 60f x x x x x

( 1 3 )x i ( 1 3 ) ( 1 3 )x i x i

2 23 3 1 9x x xi x xi i 2 2 10x x

2 4 3 2( 2 10) 3 6 2 60x x x x x x

2x

4 3 22 10x x x 3 24 2x x x

x

3 22 10x x x 26 12 60x x

6

26 12 60x x

0

4. Factor the quotient2 6x x ( 3)( 2)x x

3x 2x

ALL SOLUTIONS (remember, imaginary and its conjugate)

3, 2, 1 3 , 1 3x i i

Ex: Find the 4th degree polynomial with zeros of 1, 1, and 3i

• Turn into factors

• Remember conjugate is a factor as well!!

• Multiply and Simplify!

( 1)( 1)( 3 )x x x i ( 3 )x i

2( 2 1)x x 2( 9)x 4 3 2 22 9 18 9x x x x x

4 3 22 10 18 9x x x x

Ex: Find the cubic with zeros, 2 and 1-i where f(1)=3

1. Write the factors, including the conjugate

2. To find f(1) = 3, take the function, put a(function)=3 and solve

3. Plug in 1 for x and find a

4. Distribute a

( 2)( 1 )( 1 )x x i x i 2 2( 2)( 1 )x x x xi x xi i 2( 2)( 2 2)x x x

3 24 6 4x x x

3 24 6 4 3a x x x

3 21 4 1 6 1 4 3a 3a 3a

3 23 4 6 4x x x 3 23 12 18 12x x x