Question

Please explain in detail what is remainder theorem. If Possible, with examples please. Regards

Answer 1

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 2x2−5x−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.

We didn't need to do Long Division at all!

Example: The remainder after 2x2−5x−1 is divided by x−5

Same example as above but this time we divide by "x−5"

"c" is 5, so let us check f(5):

2(5)2−5(5)−1 = 2x25−5x5−1

= 50−25−1

= 24

The remainder is 24

Once again ... We didn't need to do Long Division to find that.

Answer 2

Let P(x) be any polynomial of degree greater than or equal to one and let 'a' be any real number.

If P(x) is divided by linear polynomial x -a, then the remainder is P(a)

for example, to find the remainder when the polynomial f(x)=x³+4x²-3x+5 is divided by x+4

solution: by remainder theorem when f(x) is divided by x+4 then remainder is f(-4)

f(-4)=(-4)³+4(-4)²-3(-4)+5

=-64+64+12+5

=17