how do you find remainder from the details provided above?If P(x) has the remainder -5 when divided by x-3 and the remainder 3 when divided by x+2, find the reminder when P(x) is divided by (x-3)(x+2)
how do you find remainder from the details provided above?If P(x) has the remainder -5 when divided by x-3 and the remainder 3 when divided by x+2, find the reminder when P(x) is divided by (x-3)(x+2)
Imagine you write the euclidean division :
There exists a polynomial Q such that :
Now let
This gives
Same thing for the division by (x+2) : we get
Now what if we consider the division by ?
There exists polynomials R and S (S is the remainder) such that :
Now once again, if you let , you get :
so we have
Similarly, if you let , you get :
So the remainder of P when divided by (x-2)(x+3) will be a polynomial S such that and
But the degree of S cannot exceed 2, since (x-2)(x+3) has a degree 2. (if S has a degree equal or superior to 2, then it still can be divided by (x-2)(x+3))
Hence we're looking for a and b in :
Now the previous working gives you the following system :
Which is very basic algebra
Thread of related interest: http://www.mathhelpforum.com/math-he...mial-help.html
By the way, the question is asking for the remainder so I don't know why you keep talking about P(x). P(x) is not the remainder.