1. ## Factors

Some help would be greatly appreciated, I'm at such a loss

2. A cubic equation has three linear factors.
in this case,two are same
hence
$\displaystyle x^3 + 3x^2 - 9x + c= (x-p)(x-p)(x-q)$
$\displaystyle x^3 + 3x^2 - 9x + c= x^3-(2p+q)x^2+(2pq+p^2)x-p^2q$
comparing the coefficients,
$\displaystyle 2p+q=-3...[1]$
$\displaystyle 2pq+p^2=-9...[2]$
$\displaystyle p^2q=-c...[3]$
we solve first two equations for p and q
from 1st equation$\displaystyle q=-3-2p$
substituting in second$\displaystyle 2p(-3-2p)=-9$
hence,after rearraging and simplifying$\displaystyle p^2+2p-3=0$
which gives$\displaystyle p=-3,1$
hence from first equation,$\displaystyle p=-3,q=3$ or $\displaystyle p=1,q=-5$
if we substitute in third equation,we get$\displaystyle c=-27,5$

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