Home | | **Maths 10th Std** | 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*)** **where*q*** **(*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***
**)** **+** ***r*_{1}** **(*x*** **)** **where
*r*_{1} (*x* ) is the new remainder. Then deg [*r*_{1}
(*x*) ] < deg [*r* (*x* )] . If the remainder *r*_{1}*
*(*x *)* *is zero, then* r *(*x*)* *is the required
GCD.

**Step 3 **If** ***r*_{1}**
**(*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*).

Find the GCD of the
polynomials**
***x*^{3}** **+** ***x*^{2}** **−** ***x*** **+** **2** **and** **2*x*^{3}** **−** **5*x*^{2}** **+** **5*x*** **−** **3** **.

Let* **f*** **(

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

Now dividing *g* (*x*)
= *x*^{3} + *x*^{2} − *x* + 2 by the new
remainder *x*^{2} − *x* +1 (leaving the constant factor), we
get

Here, we get zero
remainder

Therefore, GCD(2*x*
^{3} − 5*x* ^{2} + 5*x* − 3, *x* ^{3} + *x*
^{2} − *x* + 2) = *x* ^{2} − *x* + 1

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

Let,* **f*** **(

(*x*) =* *3*x
*^{3}* *−* *12*x *^{2}* *+* *21*x
*−* *18* *=* *3* *(*x *^{3}* *−* *4*x
*^{2}* *+* *7*x *−* *6)

Now, we shall find the
GCD of *x*^{3} − 5*x*^{2} + 10*x* − 8 and *x*^{3}
− 4*x*^{2} + 7*x* – 6

Here, we get zero as
remainder.

GCD of leading
coefficients 3 and 6 is 3.

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

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