H.C.F. of Polynomials by Long Division Method

Now we will learn how to find the H.C.F. of polynomials by long division method.

Step of the method:

(i) At first, the given expressions are to be arranged in the descending order of powers of any of its variables.

(ii) Then if any common factor is present in the terms of each expression, it should be taken out. At the time of determination of final H.C.F., the H.C.F. of these taken out factors are to be multiplied with the H.C.F. obtained by the method of division.

(iii) Like the determination of H.C.F. by the method of division in arithmetic, here also as the division is not complete, in every step the divisor of that step is to be divided by the remainder obtained. At any stage, if any common factor is present in the remainder that should be taken out, then the division in the next step becomes easier.

(iv) In every step, the term in the quotient should be found by comparing the first term of the dividend with the first term of the divisor. Sometimes, if necessary, the dividend may be multiplied by a multiplier of a factor.

1. Find the H.C.F. of 4a4 + 40a2 – 20a3 – 32a and 2a4 – 12a – 8a3 + 14a2 by using the long division method.

Solution:

(i) By arranging the two polynomials in the descending order of powers of x we get,

4a4 – 20a3 + 40a2 – 32a and 2a4 – 8a3 + 14a2 – 12a

(ii) By taking out the common factors from the terms of the expressions we get,
 4a4 – 20a3 + 40a2 – 32a = 4a(a3 – 5a2 + 10a – 8) 2a4 – 8a3 + 14a2– 12a = 2a(a3 – 4a2 + 7a – 6)

At the time of writing the final result the H.C.F. of 4a and 2a i.e. 2a is to be multiplied with the divisor of the last step.

(iii)

Therefore, the H.C.F. of 4a4 + 40a2 – 20a3 – 32a and 2a4 – 12a – 8a3 + 14a2 is 2a(a – 2)

2. Find the H.C.F. of 6m3 – 17m2 – 5m + 6, 6m3 – 5m2 – 3m + 2 and 3m3 – 7m2 + 4 by using long division method.

Solution:

It can be seen that the three expressions are arranged in the descending order of the powers of the variable ‘a’ and their terms have no common factors between them. So, by the long division method

The H.C.F. of the first two expressions is 6m2 + m – 2.

Now, it is to be seen whether the third expression is divisible by 6m2 + m – 2 or not. If it is not, then the H.C.F. of them is to be determined by the division method.
Therefore, the H.C.F. of 6m3 – 17m2 – 5m + 6, 6m3 – 5m2 – 3m + 2 and 3m3 – 7m2 + 4 is (3m + 2)