Algebra III
Lesson 46
Complex Roots – Factoring Over the
Complex Numbers
Complex Roots
Review of Complex Numbers
7-3i 2i
Which one isn’t complex?
All of them can be looked at as complex
4
a+bi a or b can = 0
Example 46.1
Write the quadratic equation with a lead coefficient of 1
whose roots are 1 + 2i and 1 – 2i.
(x – {1 + 2i})(x – {1 – 2i}) = 0
(x – 1 - 2i)(x – 1 + 2i) = 0
x2 – x + 2xi – x + 1 – 2i – 2xi + 2i – 4i2 = 0
x2 – 2x + 5 = 0
Factoring Over the Complex Numbers
This basically means to factor a polynomial.
Unfortunately these polynomials won’t factor with ordinary effort.
So, use extraordinary effort.
The quadratic equation.
To simplify the numbers factor out anything you can before using it.
Sample:
2x2 + 6
= 2(x2 + 3)
− b ± b 2 − 4ac
x=
2a
a=1 b=0 c=3
− 0 ± 0 2 − 4(1)(3)
=
2(1)
± 2 −3
=
= ± 3i
2
Factors into: 2 x − 3i x − − 3i
( ( ))( (
))
(
=
± − 12
2
)(
= 2 x − 3i x + 3i
)
Example 46.2
Factor 2x2 + 4x + 8 over the set of complex numbers.
= 2(x2 + 2x + 4)
− b ± b 2 − 4ac
x=
2a
− 2 ± 4 − 16
=
2
− 2 ± 2 2 − 4(1)(4 )
=
2(1)
− 2 ± − 12
=
2
=
− 2 ± 2 3i
2
= −1± 3i
The factoring is:
( (
))( (
))
= 2(x + 1 − 3i )(x + 1 + 3i )
2 x − − 1 + 3i x − − 1 − 3i
Example 46.3
Factor: x2 + 2x – 5
− b ± b 2 − 4ac − 2 ± 2 2 − 4(1)(− 5)
x=
=
2a
2(1)
=
− 2 ± 4 + 20
− 2 ± 24
=
2
2
−2±2 6
=
2
= −1± 6
The factoring is:
(x − (− 1 + 6 ))(x − (− 1 − 6 ))
= (x + 1 − 6 )(x + 1 + 6 )
Practice
a) Write the quadratic equation with a lead coefficient of 1 whose
roots are 2 – 3i and 2 + 3i.
(x – {2 + 3i})(x – {2 - 3i}) = 0
(x – 2 - 3i)(x – 2 + 3i) = 0
x2 – 2x + 3xi – 2x + 4 – 6i – 3xi + 6i – 9i2 = 0
x2 – 4x + 13 = 0
b) Factor x2 + 3x + 6 over the set of complex numbers.
− b ± b 2 − 4ac
− 3 ± 32 − 4(1)(6 )
x=
=
2a
2(1)
=
− 3 ± 9 − 24
− 3 ± − 15
=
2
2
− 3 ± 15i
=
2
3
15i
=− ±
2
2
⎛ ⎛ 3
15 ⎞ ⎞⎟⎛⎜ ⎛ 3
15 ⎞ ⎞⎟
⎜
⎟
⎜
⎜
i⎟ x − ⎜− −
i ⎟⎟
The factoring is: ⎜ x − ⎜ − +
2 ⎠ ⎟⎠⎜⎝ ⎝ 2
2 ⎠ ⎟⎠
⎝ ⎝ 2
⎛
3
15 ⎞⎛
3
15 ⎞
= ⎜⎜ x + −
i ⎟⎟⎜⎜ x + +
i ⎟⎟
2
2 ⎠⎝
2
2 ⎠
⎝
c) Sketch the graph of the function f(x) = (⅓)-x+2
Rewrite
y = (⅓)-x(⅓)2
1
y = 3x
9
Points:
Graph:
⎛ 1⎞
⎜ 0, ⎟
⎝ 9⎠
(2,1)
d) On the 36-mile trip to the magic fountain, Alice walked at a brisk pace.
On the way back, Alice doubled her pace. If the total trip took 6 hours,
how fast did Alice travel on the trip to the magic fountain and on the return
trip from the magic fountain?
tout + tback = 6
rt=d
t=d/r
36 36
+
=6
r 2r
⎛ 36 18
⎞
+
=
6
⎜
⎟⋅r
r
⎝ r
⎠
54 = 6r
r=
54
=9
6
rout = 9 & rback = 18