SOLUTION: Please solve: (x^2-ax+b)is a factor of x^3+3ax^2+3bx+c Show that b=-2a^2 c=-8a^3 Thank you

Question 365541: Please solve:
(x^2-ax+b)is a factor of x^3+3ax^2+3bx+c
Show that b=-2a^2

c=-8a^3

Thank you

Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
"Please solve:
(x^2-ax+b)is a factor of x^3+3ax^2+3bx+c
Show that b=-2a^2
c=-8a^3

Thank you"

let k be such that bk = c
(x^2 - ax + b)(x + k)
x^3 - ax^2 + bx + kx^2 - akx + bk
x^3 + kx^2 - ax^2 + bx - akx + bk
x^3 + (k - a)x^2 + (b - ak)x + bk

k - a = 3a
k = 4a

b - ak = 3b
-ak = 2b
ak = -2b
k = (-2b)/a

bk = c
k = c/b

4a = (-2b)/a (setting the 2 k's equal to solve for b)
4a^2 = -2b
-4a^2 = 2b
-2a^2 = b

k = (-2b)/a = (-2(-2a^2))/a (plugging b in)
k = (4a^2)/a = 4a

k = c/b (plugging k and b in to solve for c)
4a = c/(-2a^2)
4a(-2a^2) = c
-8a^3 = c

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