ax^3+bx+c has factor of form x^2+@x+1, show that a^2-c^2=ab

Question

chapter name is Relation between the Roots and co-efficients of an equation

Answer 1

Given conditions on the co-efficient show that x=0 is not a solution. also, the other factor must be linear, and, on examining co-efficient, must be of the form ax+c,

ax^3+bx+c=(x^2+px+a)(ax+c)=ax^3+(c+ap)x^2+(a+cp)x+c

Comparing co-efficient gives
c+ap=0
a+cp=b

Thus p=-c/p
So, a-c^2/a=b
a^2-c^2=ab

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