complex zeros and factored form

Question

x⁴ +10x³ -2x² +90x-99 find the complex zeros and put into factored form

Answer 1

The technique to use here is to find the real roots, use division to reduce the order of the polynomial to two, and then solve the quadratic using the quadratic formula.

 

The rational roots of x⁴ +10x³ -2x² +90x-99 are factors of 99.  These are 1, -1, 3, -3, 9, -9, 11, -11, 33, -33, 99 and -99.   Substitution reveals that 1 and -11 yield zero.  So x-1 and x+11 are factors

 

Dividing x⁴ +10x³ -2x² +90x-99 by x-1 yields  x³ + 11X² +9X +99

Dividing x³ + 11X² +9X +99 by X + 11 yields x² + 9.

 

We don't need the quadratic formula to solve  

 

X² = -9   x = +- 3i

 

Factored form (x-1)(x+11)(x-3i)(x+3i)