Remainder theorem

**Zeros of polynomials:**

The value of a polynomial at

e.g. Find the value of at

Sol. On putting in given polynomial , we get

This implies .

Thus value of

We say that a zero of a polynomial

The zeros of polynomials is obtained by equating the given polynomial

Let

Now replace

**Important observations**:

(i) Every linear polynomial has one and only one zero.

Let

then means

So

i.e. a linear linear polynomial has one and only one zero.

(ii) A zero of polynomial need not be 0.

e. g. The zeros of

(iii) 0 may be a zero of polynomial.

e.g. Take

(iv) A polynomial can have more than one zero.

**Division algorithm in polynomials:**

When we divide two numbers, we always get

Dividend =(divisor x quotient)+remainder, where

Now, let two polynomials and

{ Steps to divide a polynomial by a non-zero polynomial

- First, arrange the polynomials (dividend and divisor) in the decreasing order of its degree
- Divide the first term of the dividend by the first term of the divisor to produce the first term of the quotient
- Multiply the divisor by the first term of the quotient and subtract this product from the dividend, to get the remainder.
- This remainder is the dividend now and divisor will remain same
- Again repeat from the first step, until the degree of the new dividend is less than the degree of the divisor.}

Now

Hence where

In general , If and

In the above example the divisior is a linear polynomial . In such a situation there is a way to find the remainder called **Remainder Theorem.**

**Remainder theorem**

Let be any polynomial of degree greater than or equal to one and let

Proof. Let

since degree of degree of

In particular if

**e.g. Find the remainder when **

**is divided by .**

Sol. Here

so

=

Hence by the remainder theorem , the remainder is 2.

**Also Read: **

- Polynomials : Definition, Types of polynomials and Examples, Degree of a polynomial
- Relationship between Zeros and coefficients of a Polynomial