AoPS Academy
and 
be 2 distinct positive integers.
be in an integer. Prove why ?
with maximums 
and 
. If 
, and 
can be expressed as 
where 
is a positive integer, find .
and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
are also the roots of the equation 
. What is the third root of the second equation?
and 
? That wouldn't be a neat numbers to use in a synthetic division. Is there anyway we can do this problem WITHOUT using synthetic division?
Yes.
. From here, like you did, it's not that hard to figure it out.
?
?
are also the roots of the equation . What is the third root of the second equation?
, now use one of the roots for the second equation with synthetic division, I chose 
and then I factored the second equation to: 
.
. Third root is 
. Third root is 
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: .
symbol. Statements imply other statements. An expression on its own cannot imply anything.
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: . symbol. Statements imply other statements. An expression on its own cannot imply anything.
, now use one of the roots for the second equation with synthetic division, I chose and then I factored the second equation to: . symbol. Statements imply other statements. An expression on its own cannot imply anything. 
. What if someone used 
where they meant 
, or 
where they meant 
. It's not any different to me at least. Anyway, I'm sure your math professors won't like it either, as 10000th User hinted at.
,
.
roots. In the question it gives you the first two roots (same as the quadratic equation's) and asks for the remaining (3rd) root.
are 
.
has 3 roots because it is a cubic.
.
of the roots of the 2nd equation 
)
