By this method, we are led to determine the greatest common divisor between three or more polynomials; but they will be more simple than the proposed polynomials. It often happens, that some of the co-efficients of the arranged polynomial are monomials, or, that we may discover by simple inspection that they are prime with each other; and, in this case, we are certain that the proposed polynomials are prime with each other.

Thus, in the example of Art. 269, treated by the first method, after having suppressed the common factor a-c, which gives the results,

d2-c2 and 2ad-c2,

we know immediately that these two polynomials are prime with each other; for, since the letter a is contained in the second and not in the first, it follows from what has just been said, that the common divisor must divide the co-efficients 2d and -c2, which is evidently impossible; hence, &c.

2. We will apply this last principle to the two polynomials

and

3bcq+30mp+18bc+5mpq,

4adq-42fg+24ad-7fgq.

Since q is the only letter common to the two polynomials, which, moreover, do not contain any common monomial factors, we can arrange them with reference to this letter, and follow the ordinary rule. But as b is found in the first polynomial and not in the second, if we arrange the first with reference to b, which gives

(3cq+18c)b+30mp+5mpq,

the required greatest common divisor will be the same as that which exists between the second polynomial and the two co-efficients

3cq+18c and 30mp+5mpq.

Now the first of these co-efficients can be put under the form 3c(q+6), and the other becomes 5mp(q+6); hence q+6 is a common factor of these co-efficients. It will therefore be sufficient to ascertain whether q+6, which is a prime divisor, is a factor of the second polynomial.

Arranging this polynomial with reference to q, it becomes

(4ad-7fg)q-42fg+24ad ;

as the second part 24ad-42fg is equal to 6(4ad7fg), it follows that this polynomial is divisible by q+6, and gives the quotient 4ad-7fg. Therefore q+6 is the greatest common divisor of the proposed polynomials.

271. REMARK. It may be ascertained that q+6 is an exact divisor of the polynomial (4ad—7fg)q+24ad−42fg, by a method derived from the property proved in Art. 261.

Make q+60 or 4-6 in this polynomial; it becomes

(4ad-7fg)x-6+24ad-42fg,

which reduces to 0; hence q+6 is a divisor of this polynomial. This method may be advantageously employed in nearly all the applications of the process. It consists in this, viz. after obtaining a remainder of the first degree with reference to a, when a is the principal letter, make this remainder equal to 0, and deduce the value of a from this equation.

If this value, substituted in the remainder of the 2d degree, destroys it, then the remainder of the 1st degree, simplified Art. 68, is a common divisor.. If the remainder of the 2d degree does not reduce to 0 by this substitution, we may conclude that there is no common divisor depending upon the principal letter.

Farther, having obtained a remainder of the 2d degree with reference to a, it is not necessary to continue the operation any farther. For,

Decompose this polynomial into two factors of the 1st degree, which is done by placing it equal to 0, and resolving the resulting equation of the second degree.

When each of the values of a thus obtained, substituted in the remainder of the 3d degree, destroys it, it is a proof that the remain. der of the 2d degree, simplified, is a common divisor; when only one of the values destroys the remainder of the 3d degree, the com.

mon divisor is the factor of the 1st degree with respect to a, which corresponds to this value.

Finally, when neither of these values destroys the remainder of the 3d degree, we may conclude that there is not a common divisor depending upon the letter a.

It is here supposed that the two factors of the 1st degree with reference to a, are rational, otherwise it would be more simple to perform the division of the remainder of the 3d degree by that of the second, and when this last division cannot be performed exactly, we may be certain that there is no rational common divisor, for if there was one, it could only be of the first degree with respect to a, and should be found in the remainder of the second degree, which is contrary to hypothesis.

3. Find the greatest common divisor of the two polynomials

6x-4x4-11x3-3x2-3x-1

and

4x+2x2-18x2+3x — 5

Ans. 2x-4x2+x-1.

and

4. Find the greatest common divisor of the polynomials

20x-12x2+16x+— 15x3+14x2 — 15x+4.

15x9x+47x2-21x +28.

Ans. 5x-3x+4.

5. Find the greatest common divisor of the two polynomials

5a+b2+2a3b3+ca2 — 3a2b1+bca

and

q3+5a3d—a3b2+5aabd.

Ans. a2+ab.

Transformation of Equations..

The transformation of an equation consists in changing its form without affecting the equality of its members. The object of a transformation, is to change an equation from one form to another that is more easily resolved.

1

First Transformation.

To make the second term disappear from an equation.

272. The difficulty of resolving an equation generally dimi nishes with the number of terms involving the unknown quan, tity; thus, the equation 2=p, gives immediately =± √q, whilst the complete equation +px+q=0, requires preparation before it can be resolved.

Now, any equation being given, it can always be transformed into another, in which the second term is wanting.

For, let there be the general equation

Suppose x=u+x', u being unknown, and an indeterminate quantity; by substituting u+x' for x, we obtain

(u+x')TM+P(u+x')TM-1+Q(u+x')xTM-2+ . .' . . +T(u+x')+U=0; developing by the binomial formula,, and arranging according to the decreasing powers of u, we have

Since is entirely arbitrary, we may dispose of it in such a way

that we shall have mx'+P=0; whence x':

P

m

Substituting this

value of x in the last equation, we shall obtain an equation of the

form,

uTM+Q'uTM-2+R'um-3+

...

+T'u+U'=0.

in which the second term is wanting.

If this equation was resolved, we could obtain the values of x

corresponding to those of u, by substituting each of the values of u

P

in the equation x=u+x′, or x=u—— m

Whence we may deduce the following general rule:

In order to make the second term of an equation disappear, substitute for the unknown quantity a new unknown quantity, united with the co-efficient of the second term, taken with a contrary sign, and divided by the exponent of the degree of the equation.

Let us apply the preceding rule to the equation x2+px=q. If we take x-u-·

p.

2, it becomes (-2)2+p(u—22)=9

p2

q, or, by

performing the operations, and reducing, u2- =q, this equation

ponding values of x,

+q, consequently we obtain for, the two corres-、

273. Instead of making the second term disappear, an equation may be required, which shall be deprived of its third, fourth, &c. term; this can be obtained by placing the co-efficient of um-3, um ... equal to 0.. For example, to make the third term disappear, we make in the above transformed equation

from which we obtain two values for x', which substituted in the transformed equation reduces it to the form

u+P'um-'+R'um-3+.... T'u+U'÷0..

Beyond the third term it will be necessary to resolve equations of a degree superior to the second, to obtain the value of x': thus to cause the last term to disappear, it will be necessary to resolve the equation