1)
Fundamental Theorem of Algebra If P(x) is a polynomial function
of degree n (n > 0) with complex coefficients, then the equation P(x)
= 0 has n roots assuming you count double roots as 2, triple roots as 3,
etc. 2) Complex Conjugates
Theorem If P(x) is a polynomial function
with real coefficients, and a + bi is a solution of the equation P(x) =
0, then a - bi is also a solution. 3) If P(x) is a polynomial
with rational coefficients and a and b are also rational such that the
square root of b is irrational, then if is a root of the equation
P(x) = 0, then
is also a root. 4) If P(x) is a polynomial
of odd degree with real coefficients, then the equation P(x) = 0 has at
least one real solution. 5) For a polynomial
equation with a_{n} as the leading coefficient and a_{o}
as the constant then the following is true:

a) the sum of the roots is - a_{n-1} /a_{n}

b) the product of the roots is:

a_{o}/a_{n} if n is even

-a_{o}/a_{n} if n is odd

A useful formula to help you find equations given the root is:

x^{2} - (sum of the roots)x + (product of the roots) =
0

Sample Problems

1)
Find a quadratic equation with the root 2 + 5i
Solution:
Since complex solutions come in pairs, 2 - 5i is also a solution.
Find the sum and product and use the above formula.
Sum = ( 2 + 5i) + (2 - 5i) = 4
Product = (2 + 5i)(2 - 5i) = 4 - 25i^{2} = 4 + 25 = 29

Therefore, the equation is: x^{2} - 4x + 29 = 0 2) Find a cubic equation
with integral coefficients for 3 + i and 2.
Solution:
Again, complex solutions come in pairs. 3 - i is a solution.
Using the complex solutions, find a quadratic.
Sum = (3 + i) + (3 - i) = 6
Product = (3 + i)(3 - i) = 9 - i^{2} = 9 + 1 = 10
Therefore, the quadratic is x^{2} - 6x + 10
So the equation is: (x - 2)(x^{2} -6x + 10) = 0
x^{3} - 6x^{2} + 10x -2x^{2} + 12x - 20 = 0
x^{3} - 8x^{2} + 22x - 20 = 0
Notice that the sum of the three roots is 8 and the product is 20! 3) Find a quartic equation
with the following roots: i and 2 + i
Solution:
Again, complex come in pairs. So, -i and 2 - i are also roots
Form two quadratics for the solutions and multipy them!
Quadratic #1 sum = i + -i = 0
Product = i(-i) = -i^{2} = 1
First quadratic is: x^{2} + 1
Quadratic #2 sum = (2 + i) + (2 -i) = 4
Product = (2 + i)(2 - i) = 4 - i^{2} = 4 + 1 = 5
Second quadratic is: x^{2} -4x + 5
Therefore, the equation is: (x^{2} + 1)(x^{2} - 4x
+ 5) = 0
x^{4} - 4x^{3} + 5x^{2} + x^{2} - 4x +
5 = 0
x^{4} - 4x^{3} + 6x^{2} - 4x + 5 = 0

That's it for chapter two!! Are you ready for the sample
test?

Take me back, I don't
understand!!

Let's
go for it! I'm ready!!

Current quizaroo # 2b

1) A farmer wants to construct a rectangular
pen using the barn wall as one side of the fence. He also wants to
construct a fence down the middle of the pen parallel to the two sides.
If he has 150 feet of fencing, what is the maximum area he can enclose?

a)
25 square feet

b)
1875 square feet

c)
5625 square feet

d)
1200 square feet

e)
500 square feet

2) Find all zeros of the following
equation: 3x^{3} + 10x^{2} - x - 12 = 0
a) 3, 4, 12

b)
-1, 3, 4/3

c)
1, -3, -4/3

d)
2, 3, -4

e)
-2, -3, 4

3) Find the intersection of the following functions:
f(x) = x^{2} + 4 and g(x) = 2x

a)
(0, 0)

b)
(0,2)

c)
(-2, 4) and (1,2)

d)
They do not intersect

e)
(4, 20)

4) What are the sum
and product of the roots for the equation: 5x^{3} - 3x + 1 = 0

a) sum is 0 and product is -1/5

b) sum is 3/5 and product is -1/5

c) sum is -3/5 and product is 1/5

d) both sum and product are 0

e) sum is 1/5 and product is 3/5

5) Write a cubic function that has roots
of -3 and 2i