SOLUTION: consider the polynomial P(x)=x^3+3x^2+(3-k)x+1-k where k is a constant. a, show that P(-1)=0 for any value of k. b, find all solution to the equation P(x)=0 if k=-1

Question 870577: consider the polynomial P(x)=x^3+3x^2+(3-k)x+1-k where k is a constant.
a, show that P(-1)=0 for any value of k.
b, find all solution to the equation P(x)=0 if k=-1
Answer by math1239028(4) (Show Source): You can put this solution on YOUR website!
That's an easy one:
a. Just replace where x=-1:
P(-1)=(-1)^3+3(-1)^2+(3-k)(-1)+1-k <=>
P(-1)=-1+3-3+k+1-k=0 => P(-1)=0

b. for k=-1: P(x)=0
P(x)=x^3+3x^2+4x+2=0
And then you solve this equation:
x^3+3x^2+4x+2=0
(x+1)(x^2+2x+2)=0
x+1=0 or (x+1)^2=-1
x=-1 or x+1=i or x+1=-i
____or x=-1+i or x= -1-i
So, summing up, P(x)=0 if k=-1, for x=-1 or x=-1+i or x=-1-i.
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