+0  
 
0
81
1
avatar

A cubic polynomial p(x) has leading coefficient 1, all real coefficients, and p(3 - 4i) = 0. If p(0) = −52, find p(x).

 Jul 23, 2022
 #1
avatar+124674 
+1

Call  the  one real root , r        The other two roots are 3 + 4i  and 3  -4i

 

(x - r)  ( x - (3 +4i)) ( x - (3 -4i)   =  0

 

(x - r)  (x^2 - 6x + 25)  = 0

 

x^3 - 6x^2 + 25x   

      -  rx^2  + 6rx  - 25 r

___________________

x^3 - (6+r)x^2 + (25 + 6r)  - 25 r =    0

 

Since  p(0)  = -52

 

Then   -25r= -52    ⇒  r = 52/25 =  the real root

 

So

 

p(x)  =  x^3 - (6+52/25)x^2 + (25 +6*52/25)x - 52         simplify 

 

p(x)  =   x^3 - (202/25)x^2  + (937/25)x - 52  

 

cool cool cool

 Jul 23, 2022

7 Online Users

avatar