Question

What is the meaning of reminder theroms?

Answer 1

Remainder Theorem
If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial (x – a),then the remainder is p(a).

The Remainder Theorem
When we divide f(x) by the simple polynomial (x−c) we get:
f(x) = (x−c)·q(x) + r(x)
(x−c) is degree 1, so r(x) must have degree 0, so it is just some constant r :

f(x) = (x−c)·q(x) + r
Now see what happens when we have x equal to c:

f(c) = (c-c)*q(c)+r

f(c) = 0*q(c) +r

f(c) = r

So we get this:
The Remainder Theorem:
When we divide a polynomial f(x) by (x−c) the remainder is f(c)
So to find the remainder after dividing by x-c we don't need to do any division:
Just calculate f(c).
Let us see that in practice:
Example: The remainder after 2x^2−5x−1 is divided by x−3
(Our example from above)
We don't need to divide by (x−3) ... just calculate f(3):
2(3)^2−5(3)−1 = 2x9−5x3−1
= 18−15−1
= 2
And that is the remainder we got from our calculations above.