Remainder Theorem:

Remainder theorem states that when a polynomial function f(x) is divided by (x –a) where a is a constant, it will give us a remainder f(a).

Remainder of Let us understand this with the help of some examples.

Example: Find the remainder of Solution: Solution:

Method 1:

Here, note that numerator is in the form of powers of 2 hence we have to convert the denominator also in the form of powers of 2. Here denominator 15 = 24 –1

Now denominator is in the form of 24, so we can  write numerator also in the form of 24.

2120 =  (24)30. Now if take    24 = x

then numerator => x30

denominator = x –1.

Now by applying remainder theorem concept

a = 1 so remainder is f(a) =  (1)30 = 1.

Chinese Remainder Theorem:

Chinese theorem is used when we have a big composite number in denominator and complex power in numerator.

Let us understand this with the help of some examples:

Case 1: When remainder obtained by both numbers is equal

Example: Find the remainder of Solution:

Step 1:

Write the denominator as product of two coprimes

In this case 15 = 5 * 3

And dedude remainder for respective factor

Step 2: Case 2: When remainders obtained by both numbers are unequal

Example: Find the remainder of Solution:

Step 1: Step 2:

Now we need values of a and b for that N is same  Try out some more questions based on this concept so as to get a good hold of this topic.

EXERCISE:

Question 1. Find the remainder of .

(1) 22

(2) -22

(3) 11

(4) 0

Solution: Option 2

Here given function is f(x) = 3×3 –3x –4, and it is divided by (x + 2) hence here a = –2, so remainder of this division if f(–2) = 3 × (–2)3 –3 × (–2) –4 = –24 + 6–4 = –22

Question 2. Find remainder of .

(1) 12

(2) 32

(3) 162

(4) 0

Solution: Option 3

Step 1:

Split 100 into two coprimes

Since 100 = 25 × 4 hence we will 1st find remainder of

Step 2:  Question 3. Find remainder of (1) 37

(2) 52

(3) 16

(4) 61

Solution: Option 4

Step 1 :

Split 100 into two coprimes

Since 100 = 25 × 4 hence we will 1st find remainder of

Step 2:- Step 3 :

We need to find value of a for that     