When asked to factorise say P(z)=z3-5z2+8z-6 and i know z-3 is a factor, how do you find the quadratic bit of it without doing long division. I remember there was a way by just inspection, something like looking at the last term of the original p(z) but i forgot it. Can someone explain it to me?
Thanks
Not sure, but you might be thinking about how the three terms in brackets need to multiply together to make the last term.
(say you had (x-3)(x-2)(x+1) you know the last term of the expanded polynomial will be 6 as -3 x -2 x 1 = 6)
Meaning the remaining two must multiple to make 2 (as we already have -3 and 2 x -3 = -6).
So something like
and
(as
) could work, and does.
edit: Oh I misinterpreted your question, my apologies.
You have;
You know
is a factor
We know that the coefficient of the
term is
, which is calculated by
therefore
We also know that the last term in the expanded polynomial is
, which will be -
, thus
Lastly the coefficient of the
is
, which is calculated by the addition of
and
, therefore
and hence;
Giving our desired roots of