Show that if we replace x by [λ-(a/3)] and simplify, we end up with an equation that doesn't have a λ^2 term - that is, an equation in the form λ^3+pλ+q=0

(Given the general cubic equation with rational coefficients: x^3 + ax^2 + bx + c)

Question

Show that if we replace x by [λ-(a/3)] and simplify, we end up with an equation that doesn't have a λ^2 term - that is, an equation in the form λ^3+pλ+q=0

(Given the general cubic equation with rational coefficients: x^3 + ax^2 + bx + c)

anonymous · Answer

Plug that term in for x in all those places and show what you get

anonymous · Answer

i got this when i expanded it all

anonymous · Answer

Okay now combine like terms

anonymous · Answer

but i found nothing to combine

anonymous · Answer

You should recheck your expansion then because a lot of that should clear out

anonymous · Answer

can you help me i dont know what i did wrong

phi · Answer

It looks ok, as far as you went.

phi · Answer

the 2nd and 3rd terms combine.  there are others.  match exponents on lambda and a

anonymous · Answer

ohhhhh wow im silly. i totally overlooked some of them. thank you!

phi · Answer

and I assume you will fix your bx  (you changed a lambda into an x)

anonymous · Answer

yeah

phi · Answer

and you should write
$$ -2a \lambda \frac{a}{3} $$
as

$$ -2 \lambda \frac{a^2}{3} $$
so it's easier to find matching terms

anonymous · Answer

this is what i simplified so far... im not sure what to do about the b and the ab terms though or maybe i made a mistake before?

phi · Answer

you should group them in powers of lambda
so that you match the coefficients with
λ^3+pλ+q=0

in other words,  p and q are going to be some ugly combinations of a and b

phi · Answer

your -a^2 λ  should be divided by 3 ?

also, you are missing the C term
and the sign on  bλ  should be + ?

anonymous · Answer

thanks for those catches! i got down to this, does it seem to be correct?

phi · Answer

you want 27 C/27   
and I got (3b-a^2)/3  for p

phi · Answer

though looking at it more carefully, your q is OK.  (but I still would put 27C / 27 rather than have c dangling off to the side.)

anonymous · Answer

ok thank you so much for all your help!

anonymous · Answer

but how do you get (3b-a^2)/3 ... i can only get it with +

anonymous · Answer

oh never mind i got it

anonymous · Answer

Would you be able to help me with another problem that's related to this one?

anonymous · Answer

yep

anonymous · Answer

using the procedure from part 1, depress the equation x^3+6x^2+9x+4=0