Fundamental Theorem of Algebra

Fundamental Theorem of Algebra - a polynomial function of degree "n" and has "n" zeroes

ex.      f(x) = x + 4    exponent of one means it has one solution/x-intercept/zero

ex.      f(x) = x²+ x - 12   exponent of two means it has two solutions/x-intercepts/zeroes

 

ex.      Solve the equation x⁵ + x³ + 2x² - 12x + 8 using synthetic division.

           a)  Use calculator to check for zeroes

                 

x = -2, 1, 1 (1 has a multiplicity of 2 which is identified by a bounce on the graph)

           b)  Use synthetic division with the found zeroes

                    -2     1     0     1     2    -12    8

                                    -2    4   -10   16    -8

                      1     1     -2    5   -8      4      0       x⁴ - 2x³ + 5x² - 8x + 4

                                     1    -1    4    -4 

                      1      1     -1     4   -4     0              x³ - x² + 4x - 4

                                      1     0    4

                              1      0     4    0                      x² + 4

           c)  Since the equation has an exponent of 5 and we know that three are real, two must be imaginary.

                Solve the resulting function (result of synthetic division).

                           x² + 4 = 0      (set equal to zero)

                                 x² = -4     (get "x" by itself)

                                   x = + 2i  (square root and simplify)

****Note: Imaginary roots always come in conjugate pairs.

                                     ex.       If one solution is 1 + 7i, you know that another solution is 1 - 7i.

           ex.     Solve f(x)= x⁴ - 3x³ + 6x² + 2x - 60 if one root is 1 + 3i.

                      a)  Knowing that 1 + 3i is a solution, you also know that 1 - 3i is also a solution, thus leaving two more root to find.

                      b)  Use synthetic division with the given zeroes.

                            1 + 3i     1        -3          6          2          -60

                                                   1+3i     -11-3i    4-18i       60

                            1 - 3i      1      -2+3i     -5-3i    6-18i         0

                                                    1-3i     -1+3i   -6+18i

                                           1        -1          -6        0                     x² - x - 6

                      c)  Since the equation has an exponent of 4 and we know two imaginary roots, we must find the remaining two.

                           Solve the resulting function (result of synthetic division)

                                      x² - x - 6 = 0          (set equal to zero)

                                      (x - 3)(x + 2) = 0    (factor)

                                            x = 3, -2          (set equal to zero and solve)