Section 2-6:  General Theorems for Polynomials
 
Try the quiz at the bottom of the page!
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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 an as the leading coefficient and ao as the constant then the following is true:
 
a)  the sum of the roots is - an-1 /an
b)  the product of the roots is:
ao/an if n is even
-ao/an if n is odd

 
A useful formula to help you find equations given the root is:
 
x2 - (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 - 25i2 = 4 + 25 = 29

                    Therefore, the equation is:  x2 - 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 - i2 = 9 + 1 = 10
 
                    Therefore, the quadratic is x2 - 6x + 10
 
                    So the equation is:  (x - 2)(x2 -6x + 10) = 0
                                                     x3 - 6x2 + 10x -2x2 + 12x - 20 = 0
                                                     x3 - 8x2 + 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) = -i2 = 1
                        First quadratic is:  x2 + 1
 
                    Quadratic #2    sum = (2 + i) + (2 -i) = 4
                                       Product = (2 + i)(2 - i) = 4 - i2 = 4 + 1 = 5
                        Second quadratic is: x2 -4x + 5
 
            Therefore, the equation is:  (x2 + 1)(x2 - 4x + 5) = 0
                                                           x4 - 4x3 + 5x2 + x2 - 4x + 5 = 0
                                                           x4 - 4x3 + 6x2 - 4x + 5 = 0


 
That's it for chapter two!!  Are you ready for the sample test?
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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:  3x3 + 10x2 - 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) = x2 + 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:  5x3 - 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
  • a)  x2 + 3x + 1 = 0
  • b)  x3 + 3x2 + 4x + 12 = 0
  • c)  x3 - 3x2 + 4x - 12 = 0
  • d)  x3 + 4x2 + 3x + 12 = 0
  • e)  x3 - 3x2 - 4x + 12 = 0
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