Remainder theorem

**Zeros of polynomials:**

The value of a polynomial

e.g. Find the value of at .

Sol. On putting in given polynomial , we get

This implies .

Thus value of at is 12.

We say that a zero of a polynomial is a number such that . It means is zero of a polynomial if the value of that polynomial at is zero.

The zeros of polynomials is obtained by equating the given polynomial . We say is a polynomial equation and is root of the polynomial equation .

Let be a constant polynomial, then .

Now replace with any number we still get . This implies constant polynomials has no zeros. In case of zero polynomial , every real number is a zero of the zero polynomial.

**Important observations**:

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

Let be a linear polynomial,

then means .

So is the only zero of .

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 are -2 and 2.

(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 . When remainder becomes zero, we say divisor and quotient both are factors of dividend.

Now, let two polynomials and . Divide by

{ 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 are two polynomials such that and , then we can find a polynomial as quotient and as remainder, where or .

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 be any real number. If is divided by the linear polynomial , then remainder is .

Proof. Let be any polynomial of degree greater than or equal to 1. Suppose is divided by then by using division algorithm theorem , can be written as

since degree of degree of , this implies degree of =0 ( degree of )

(a constant polynomial)

In particular if then which proves the theorem.

**e.g. Find the remainder when is divided by .**

Sol. Here and zeros of is 1.

so

=

Hence by the remainder theorem , the remainder is 2.

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