Given that 3−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable. f(x)=x4−2x3−4x2+130x−125

Question

Given that 3−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

f(x)=x^4−2x^3−4x^2+130x−125

Answer 1

Since 3- 4i  is a zero, by the conjugate root theorem, so must 3 + 4 i be a zero.   

This implies that f(x) has the form

f(x) = [x -(3 -4i) ][x - (3 + 4i)]  Q(x)   where Q is a second order polynomial

 

multiplying things out:   f(x) = (x² -6x +25) Q(x).   

 

This means that Q(x) =  (x⁴ - 2 x³ - 4 x² +130 x -125)/(x² -6 x +25)

 

Either long division or synthetic division can be used to work out Q

The result is   Q(x) = x² + 4x -5  = (x + 5)_ (x -1)

The complete factorization (in terms of real coefficients) is

 f(x) = (x² - 6 x +25 )( x +5) (x -1)

 

 

Answer 2

 

3+4i is also a zero.

 

( x  - (3+4i)) ( x - (3-4i)) =

(x - 3 - 4i)) (x - 3 + 4i)) =

 

(x-3)^2 + 16  = x^2 - 6x + 25

 

 

Synthetic division says:

                    1      +4    -5

             ________________________________

1 -6 25  |      1         -2       -4     130    -125

                    1        -6       25

                  --------------------------------------

                              4    -29       130

                              4    -24       100

                             --------------------

                                       -5       30   -125

 

 

x^2 + 4x - 5 is the quotient

 

 

 

(x  +  5)( x  -     1 ) are the other two factors

 

x=-5 and x=1 are also zeros

 

(x + 5)(x - 1) ( x - 3  - 4i) ( x - 3 + 4i) are the factors

the zeros are x=-5, x=1, x = 3 + 4i, x = 3 - 4i