## Division Algorithm for Polynomials

Let and be polynomials of degree n and m respectively such that m £ n. Then there exist unique polynomials and where is either zero polynomial or degree of degree of such that .

is dividend, is divisor.

is quotient, is remainder.

## Solved Examples based on Division Algorithm for Polynomials

### Example:

Apply the division algorithm to find the quotient and remainder on dividing by as given below.

,

### Solution:

\ Þ

### Example:

Check whether the first polynomial is a factor of 2nd polynomial by applying division algorithm.

,

### Solution:

\ First polynomial is a factor of second polynomial.

### Example:

If is the factor of the polynomial prove that and

### Solution:

Let

Since is a factor of

Þ Þ

Þ Þ

Þ

as

\

### Example:

What must be subtracted from so that the resulting polynomial is exactly divisible by

### Solution:

By division algorithm, we have

Dividend = Divisor ´ Quotient + Remainder

Þ Dividend – Remainder = Divisor ´ Quotient

Þ

On dividing by we get

Thus if we subtract from it will be divisible by

### Example:

What must be added to so that the resulting polynomial is divisible by

### Solution:

By division algorithm, we have

Þ

Þ

Clearly RHS is divisible by

\ LHS is also divisible by

Thus, if is added to then the resulting polynomial becomes divisible by

Hence, we should add so that the resulting polynomial is divisible by g(x).

### Example:

If two zeros of the polynomial are find other zeros.

### Solution:

and are zeros of

\

=

= is a factor of .

Let us divide by

Hence, other two zeros of are zeros of polynomial

=

=

=

=

Other two zeros are –5, 7

### Example:

On dividing a polynomial by a polynomial the quotient and remainder are and respectively. Find

### Solution:

=

\

### Example:

Find all the zeros of if you know that two of its zeros are and

### Solution:

Given two zeros : and

Þ Quadratic polynomial =

Þ is a factor of

Applying the division algorithm theorem to given polynomial

and

Clearly, we have

Therefore, all the zeros are : and

Þ other two zeros are :