Polynomial Equation

Question

i and -2i are two zeroes of p(x)  = x⁶ -2x⁵ +2x⁴ -10x³ -11x² -8x -12. Find the other zeroes and write p(x) as a product of linear factors. 

Answer 1

Based on the zeros you have provided, the two factors that we know already are (x² + 1) and (x² + 4).  Multiply then together to obtain

 

(x⁴ + 5x² + 4).

 

 

Next, we will divide p(x) by this factor we obtained.

 

 

                                                              x² - 2x - 3

                      _____________________________________

(x⁴ + 5x² + 4) | x⁶ - 2x⁵ + 2x⁴ - 10x³ - 11x² - 8x - 12

                        x⁶             5x⁴              4x²

                        __________________________

                            -2x⁵   - 3x⁴ - 10x³ - 15x² - 8x

                            -2x⁵            - 10x³          - 8x

                          _____________________________

                                     -3x⁴           - 15x²        - 12

                                     -3x⁴           -15x²        - 12

                                      ________________________

                                                                    0

 

 

Since the remainder is zero, p(x) in linear factored form is

 

p(x) = (x - i)(x + i)(x - 2i)(x + 2i)(x - 3)(x + 1)

Answer 2

Complex roots come in conjugate pairs:

i and -i, -2i and 2i.

Four of the factors are"

(x - i)(x + i)(x - 2i)(x + 2i)

Use synthetic substitution/division for the other two factors.