Division of Polynomial by another Polynomial Long Division


 
 
Concept Explanation
 

Division of Polynomial by another Polynomial Long Division

Division of a polynomial by a binomial by using long division

For dividing a polynomial by a binomial, we may follow the following steps:

Step I   Arrange the terms of the dividend and divisor in descending order of their degrees.

Step II  Divide the first term of the dividend by the first term of the divisor to obtain the first term of the

            quotient.

Step III  Multiply the divisor by the first term of the quotient and subtract the result from the dividend to

             obtain the remainder.

Step IV  Consider the remainder  (if any)  as dividend and repeat step II to obtain the second term of

             quotient.

Step V   Repeat the above process till we obtain a remainder which is either zero or a polynomial of

             degree less than that of the divisor.

Example 1    Divide large 6+x-4x^{2} + x^{3};;by;;x-3.

Solution         We go through the following steps to perform the division:

Step I            We write the terms of the dividend as well as of divisor in descending order of their

                     degrees. Thus, we write

                     large dpi{120} large 6+x-4x^{2}+x^{3};;as;;x^{3}-4x^{2}+x+6;;and;;x-3;;as;;x-3

Step II           We divide the first term large x^{3} of the dividend by the first term large x of the divisor and obtain

                     large frac{x^{3}}{x}=x^{2} as the first term of the quotient.

Step III          We multiply the divisor large x-3  by the first term large x^2 we get large x^3-3x^2   of the quotient

                     and subtract the result from the dividend

                    large x^{3}-4x^{2}+x+6.

                   We obtain large -x^{2}+x+6  as the remainder.

Step IV        We take large -x^{2}+x+6 as the new dividend and repeat step II to obtain the second

                   term large (-frac{x^{2}}{x}=)-x of the quotient.

Step V         We multiply the divisor large x-3 by the second term large -x of the quotient and subtract the

                   result large -x^{2} + 3x from the new dividend. We obtain large -2x + 6 as the remainder.

Step VI        Now we treat large -2x + 6 as the new dividend and divide its first term large -2x by the first

                   term large x of the divisor to obtain large frac{-2x}{x}=-2 as the third term of the quotient.

Step VII      We multiply the divisor large x-3 and the third term large -2 of the quotiemt and subtract the

                  result large -2x+6 from the new dividend. We obtain large 0 as the remainder.

                  Thus, we can say that

                          large (6+x-4x^{2}+x^{3})div (x-3)=x^{2}-x-2

                  or,                large frac{6+x-4x^{2}+x^{3}}{x-3}=x^{2}-x-2

                                

Sample Questions
(More Questions for each concept available in Login)
Question : 1

If the degree of the dividend is 5 and the degree of the divisor is 3, then the degree of the quotient will be ________________-

Right Option : D
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Explanation
Question : 2

If  x^{4}+3x^{2}+7 is divided by 3x+5, then the possible degrees of quotient and remainder are ___________________

Right Option : A
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Explanation
Question : 3

If x^{5}+2x^{4}+x+6  is divided by g(x) and quotient is x^{2}+5x+7, then the possible degree of remainder is  _____________________

Right Option : C
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Explanation
 
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