Remainder Theorem


It states that the remainder of the division of a polynomial f(x) by a linear polynomial x-a is equal to f(a) In particular, x-a is a divisor of f(x) if and only if f(a)=0.

We can divide polynomials.

f(x) ÷ g(x) = q(x) with a remainder of r(x)

Using long division method, 2x2-5x-1 is  divided by x-3

f(x) is 2x2-5x-1

g(x) is x-3

After dividing we get the answer 2x+1, but there is a remainder of 2.

q(x) is 2x+1

r(x) is 2

In the style f(x) = g(x)·q(x) + r(x) we can write:

2x2−5x−1 = (x−3)(2x+1) + 2

When we divide by a polynomial of degree 1 (such as "x−3") the remainder will have degree 0 (in other words a constant, like "4").

Using the Remainder Theorem, when we divide a polynomial f(x) by x-c we get:

f(x) = (x−c)·q(x) + r(x)

But r(x) is simply the 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 r equals f(c)

So when we want to know the remainder after dividing by x-c we don't need to do any division, we need to just calculate f(c).

Example: 2x²−5x−1 divided by x-3

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