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Example, Solution | Algebra - Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of Polynomials | 10th Mathematics : Algebra

Chapter: 10th Mathematics : Algebra

Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of Polynomials

The following procedure gives a systematic way of finding Greatest Common Divisor of two given polynomials f (x ) and g (x) .

Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of Polynomials

In our previous class we have learnt how to find the GCD (HCF) of second degree and third degree expressions by the method of factorization. Now we shall learn how to find the GCD of the given polynomials by the method of long division.

As discussed in Chapter 2, (Numbers and Sequences) to find GCD of two positive integers using Euclidean Algorithm, similar techniques can be employed for two given polynomials also.

The following procedure gives a systematic way of finding Greatest Common Divisor of two given polynomials f (x ) and g (x) .

Step 1 First, divide f(x) by g (x) to obtain f (x ) = g (x )q (x ) + r (x) whereq (x) is the quotient and r (x) is the remainder. Then, deg [r (x)] < deg [g (x )]

Step 2 If the remainder r (x) is non-zero, divide g (x) by r (x) to obtain g (x ) = r (x )q (x ) + r1 (x ) where r1 (x ) is the new remainder. Then deg [r1 (x) ] < deg [r (x )] . If the remainder r1 (x ) is zero, then r (x) is the required GCD.

Step 3 If r1 (x ) is non-zero, then continue the process until we get zero as remainder. The divisor at this stage will be the required GCD.

We write GCD [f (x ), g(x)]  to denote the GCD of the polynomials f (x ), g (x ).

Note

If f (x ) and g (x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend. In case, if both have the same coefficient then compare the next least degree’s coefficient and proceed with the division.

Progress Check

1. When two polynomials of same degree has to be divided, __________ should be considered to fix the dividend and divisor.

2. If r (x ) = 0 when f(x) is divided by g(x) then g(x) is called ________ of the polynomials.

3. If f (x ) = g (x )q (x ) + r (x), _________ must be added to f(x) to make f(x) completely divisible by g(x).

4. If f (x ) = g (x )q (x ) + r (x), _________ must be subtracted to f(x) to make f(x) completely divisible by g(x).

 

Example 3.10

Find the GCD of the polynomials x3 + x2 x + 2 and 2x3 5x2 + 5x 3 .

Solution

Let f (x ) = 2x 3 5x 2  + 5x 3 and g (x) = x 3 + x 2x + 2


− 7(x 2x + 1) ≠ 0 , note that -7 is not a divisor of g (x)

Now dividing g (x) = x3 + x2x + 2 by the new remainder x2x +1 (leaving the constant factor), we get


Here, we get zero remainder

Therefore, GCD(2x 3 − 5x 2 + 5x − 3, x 3 + x 2x + 2) = x 2x + 1

 

Example 3.11

Find the GCD of 6x 3 30x 2 + 60x 48 and 3x 3 12x 2 + 21x −18.

Solution

Let, f (x) = 6x 3 30x 2 + 60x 48 = 6(x 3 5x 2 + 10x 8) and g

(x) = 3x 3 12x 2 + 21x 18 = 3 (x 3 4x 2 + 7x 6)

Now, we shall find the GCD of x3 − 5x2 + 10x − 8 and x3  − 4x2 + 7x – 6


Here, we get zero as remainder.

GCD of leading coefficients 3 and 6 is 3.

Thus, GCD [(6x 3 − 30x 2  + 60x − 48, 3x 3 − 12x 2  + 21x − 18)] = 3(x −2) .

 

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