Hello, maybe you are looking for this?: http://www.mathhelpforum.com/math-he...lynomials.html
I know how to use both the remainder theorem and the factor theorem, but I'm not sure how to use them in relation to this problem:
When a polynomial is divided by , the remainder is .Any help would be greatly appreciated, even if it's just a few hints to point me in the right direction.
When a polynomial is divided by , the remainder is .
Find the remainder when the polynomial is divided by .
Hello, maybe you are looking for this?: http://www.mathhelpforum.com/math-he...lynomials.html
I can't think of a less complex method to use. Perhaps you just need a little more explanation red_dog's method.
You are given a polynomial f(x). You know that when you divide f(x) by x - p the remainder is . So you know that
by the division theorem, where is some polynomial. By the remainder theorem (or just plain substitution) we also see that
Now, f(x) divided by x - q leaves a remainder of . So by exactly the same line of reasoning we have that
where is another polynomial.
Now we wish to divide f(x) by (x - p)(x - q). Now, we are dividing by a quadratic polynomial. When we do this in general we are not left with a constant remainder, we are left with a remainder polynomial of one degree less than the divisor: we are left with a first degree polynomial as the remainder. So f(x) must be of the form:
where is another polynomial.
Possibly you would be more comfortable with this format?
This is the same statement.
Plugging x = p and x = q into this equation gives us:
and
But we already know that and so we have the system of equations:
Now solve for a and b:
(Note that if we have no solution for a and b, so the problem really needed so specify that .)
So when f(x) is divided by (x - p)(x - q) we get a remainder of .
If you want a concrete example of all this, consider and p = -1 and q = 2.
-Dan