CBSE Tutorials

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 :