Warm Up. Find all
zeros. Graph.
x  2x  7x  4
3
2
Warm Up. Find all
zeros. Graph.
x  2x  7x  4
3
2
Possible Factors :  1,  2,  4
11 2 7 4
1
3 4
1 3 4 0
x  1x 2  3x  4
x  1x  4x  1
x 1
x  4
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.
x  3x  2 0
x  3 x  2
Goes through both!
x  x6
2
Find all zeros.
x  4x  4
2
Find all zeros.
x  4x  4
2
x  2x  2 0
x  2
Multiplicity of 2.
Just Touches.
Find all zeros.
x  4x
3
Find all zeros.


x x 4 0
2
x0
x 40
2
x  4
x  2i
2
x  4x
3
Find all zeros.
x  16
4
Find all zeros.
x
2


 4 x2  4  0
x  16
4
x  2x  2x2  4  0
x  2 x  2 x  2i x  2i
Find all zeros.
x  3 x  4 x  12
3
2
Find all zeros.
x  3 x  4 x  12
3
x x  3  4x  3
2
x
2

 4 x  3
x  2i x  2i x  3
2
Find all zeros.
x  4x  4x  4
3
2
Find all zeros.
x  4x  4x  4
3
2
Possible Factors :  1,  2,  4
Using the remainder theorem we
see that there are NO real solutions!
Find all zeros.
6x  5x 1
4
3
6x  5x 1
4
Find all zeros.
3
1
1
1
1
Possible Factors :
 1,  ,  , 
 1,  2,  3,  6
2
3 6
Synthetic division gets you to the following:

1 2

x  1 x   6 x  2 x  2
2

x  1
1
x
2



2 3x 2  x  1  0
Now use the
Quadratic Formula
for this part.
x
 1  1  431
23
 1   11
6
 1  i 11
x
6
x
Find all zeros.
x  x  2 x  12 x  8
5
3
2
Find all zeros.
x  x  2 x  12 x  8
5
3
2
Possible Factors :  1,  2,  4,  8
Synthetic Division gives
x  1x  1x  2x
2

4 0
x  1 x  2 x  2i x  2i
Multiplicity of 2
Find all zeros.
x  6 x  27 x  20
4
3
Find all zeros.
x  6 x  27 x  20
4
3
Possible Factors :  1,  2,  4,  5,  10,  20
Synthetic Division gives
x  1x  4x
2
 3x  5

Quadratic Formula completes it.
 3  29
x  1 x  4 x 
2
Complex Conjugates
Complex Zeros

Always come in conjugate pairs
3  2i
3  2i
i
3i
i
 3i
Write a polynomial with the given zeros: 2, i, -i
x  2 x  i x  i
x  2x  i x  i 
2
2
x  2x  i 
x  2x 2  1
x3  x  2 x 2  2
x3  2 x 2  x  2
Write a polynomial with the given zeros: 3, 4i
x  3 x  4i x  4i
Remember – they come in pairs!
x  3x  4i x  4i 
2
2
x  3x  16i 
x  3x 2  16
x  3 x  16 x  48
3
2
Write a polynomial with the given zeros: 4, 2-3i
x  4 x  2  3i x  2  3i
x  4x  2  3i x  2  3i 
x  4x 2  2 x  3xi  2 x  4  6i  3xi  6i  9i 2 
x  4x 2  4 x  13
x 3  4 x 2  13x  4 x 2  16 x  52
x 3  8 x 2  29 x  52
Find all zeros, given that (2i) is a root.
f ( x)  x  2 x  x  8x  12
4
2i
 2i
x2  2x  3  0
x  3x  1  0
1
3
2
1
2
8
 12
2i 4i  4  6i  8 12
1 2  2i 4i  3  6i
 2i  4i
6i
0
1 2 3
x  2i x  2i x  3 x  1
Find all zeros, given that (3 – 5i) is a
solution.
f ( x)  x  7 x  40 x  34
3
2
Find all zeros, given that (3 – 5i) is a
solution.
f ( x)  x  7 x  40 x  34
3
2