What is remainder theorem

What is remainder theorem?

Table of contents
1. What is remainder theorem?
2. Examples of remainder theorem

From the remainder theorem we can find the remainder without dividing any one polynomial.

When we divide a polynomial, we can write it as:

We divide 2x2 - 5x - 1 by x - 3:

As you have seen, we got 2x + 1 answer or quotient by dividing it but with this we got 2 remainder as well.

q (x) = 2x + 1

r (x) = 2

We call it f (x) = d (x).  Q (x) + r (x) can be written as:

2x2 - 5x - 1 = (x - 3) (2x + 1) + 2

When we divide f (x) by a simple polynomial x - c, we get:

f (x) = (x - c).  q (x) + r (x)

As we know that the degree of x - c is 1, the degree of r (x) must be 0 so that r will become a constant.

f (x) = (x - c).  q (x) + r

Now we will see what will happen by keeping r equal to c:

f (c) = (c - c).  q (c) + r

f (c) = (0).  q (c) + r

f (c) = r

As we can see when we divide a polynomial f (x) by x - c, the remainder is f (c).

So now we know that if we want to find the remainder, then we don't really need to divide it, we just have to find f (c) because it is equal to the remainder.

Other examples:

Find the remainder of a polynomial 2x2 - 5x - 1 without x - 3:

As we saw in the above process, to find the remainder, we do not really have to divide by x - 3, just we have to find f (3) and the remainder will be equal to that.

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

To find f (3), we have to place 3 in the whole polynomial instead of x and solve.

f (3) = 2 (3) 2 - 5 (3) - 1

= 2 * 9 - 5 * 3 - 1

= 18 -15 -1

= 2

So we have taken its remainder without dividing it which is 2. So as you must have understood, we do not have to actually divide to get the remainder, it can also be found without dividing it.

Find the remainder of a polynomial 2x2 - 5x - 1 without x - 5 divided by:

As you can see, we are given a polynomial and we have to find the remainder without dividing it.  As we saw in the above process, we can also remove the remainder without dividing it.  For that we just have to extract f (c).  So let's draw f (c) of this polynomial.

f (x) = 2x2 - 5x - 1, c = 5

As we can see above, c is equal to 5.  So now we have to put 5 in place of all x in the polynomial to find c, this will bring f (c).

f (5) = 2 (5) 2 - 5 (5) - 1 = 2 * 25 - 5 * 5 -1

= 50 - 25 -1

= 24

So as we saw that by placing c in place of x, we got 24 which is f (c).  As we know that f (c) equals the remainder.  Hence the remainder will be 24.  One more time we have removed the remainder without parting.