2. What is the equation whose roots are 1, 2, and 3?

Ans. x37x+6= 0.

3. What is the equation whose roots are 3,

4. What is the equation whose roots are 3+5, 3-5,

and

- 6?

Ans. 13 32x24 0.

5. What is the equation whose roots are 1, and - 6?

Ans. x6+3x5 41x4 87x3400x2 + 444x

6. What is the equation whose roots are 2-1, and -3?

7200.

Ans. x3 - x2 - 7x + 15 = 0

Greatest Common Divisor.

252. The principle of the greatest common divisor is of frequent application in discussing the nature and properties of equations, and before proceeding further, it is necessary to investigate a rule for determining the greatest common divisor of two or more polynomials.

The greatest common divisor of two or more polynomials is the greatest algebraic expression, with respect both to co-efficients and exponents, that will exactly divide them.

A polynomial is prime, when no other expression except 1 will exactly divide it.

Two polynomials are prime with respect to each other, when they have no common factor except 1.

253. Let A and B designate any two polynomials arranged with reference to the same leading letter, and suppose the polynomial A to contain the highest exponent of the leading letter. Denote the greatest common divisor of A and B by D, and let the quotients found by dividing each polynomial by D,

be represented by A' and B' respectively. We shall then have

whence,

A = AX D and B = B' x D.

Now, D contains all the factors common to A and B. For, if it does not, let us suppose that A and B have a common factor d which does not enter D, and let us designate the quotients of A' and B', by this factor, by A" and B". We shall then have,

A = A". d. D and BB".d.D;

Since A" and B" are entire, both A and B are divisible by d. D, which must be greater than D, either with respect to its co-efficients or its exponents; but this is absurd, since, by hypothesis, D is the greatest common divisor of A and B. Therefore, D contains all the factors common to A and B.

Nor can D contain any factor which is not common to A and B. For, suppose D to have a factor d' which is not con tained in A and B, and designate the other factor of D by D'; we shall have the equations,

A = A'.d'. D' and BB'.d'. D';

or, dividing both members of these equations by d',

Now, the second members of these two equations being entire, the first members must also be entire; that is, both A and B are divisible by d', and therefore the supposition that d' is not a common factor of A and B is absurd. Hence,

1st. The greatest common divisor of two polynomials contains all the factors common to the polynomials, and does not contain any other factors.

254. If, now, we apply the rule for dividing A by B, and continue the process till the greatest exponent of the leading letter in the remainder is at least one less than it is in the polynomial B, and if we designate the remainder by R, and the quotient found, by Q, we shall have,

If, as before, we designate the greatest common divisor of A and B by D, and divide both members of the last equation by it, we shall have,

Now, the first member of this equation is an entire quantity,

R

and so is the first term of the second member; hence D must be entire; which proves that the greatest common divisor of A and B also divides R.

If we designate the greatest common divisor of B and R by D', and divide both members of equation (1) by it, we shall have,

Now, since by hypothesis D' is a common divisor of B and R, both terms of the second member of this equation are entire; hence, the first member must be entire; which proves that the greatest common divisor of B and R, also divides A.

We see that D', the greatest common divisor of B and R cannot be less than D, since D divides both B and R; nor can D, the greatest common divisor of A and B, be less than D', because D' divides both A and B; and since neither can be less than the other, they must be equal; that is, D = D'. Hence,

2d. The greatest common divisor of two polynomials, is the same as that between the second polynomial and their remainder after division.

From the principle demonstrated in Art. 253, we see that wo may multiply or divide one polynomial by any factor that is

not contained in the other, without affecting their greatest common divisor.

255, From the principles of the two preceding articles, we deduce, for finding the greatest common divisor of two polynomials, the following

RULE.

1. Suppress the monomial factors common to all the terms of the first polynomial; do the same with the second polynomial; and if the factors so suppressed have a common divisor, set it aside, as forming a factor of the common divisor sought.

II. Prepare the first polynomial in such a manner that its first term shall be divisible by the first term of the second polynomial, both being arranged with reference to the same letter: Apply the rule for division, and continue the process till the greatest exponent of the leading letter in the remainder is at least one less than it is in the second polynomial. Suppress, in this remainder, all the factors that are common to the co-efficients of the different powers of the leading letter; then take this result as a divisor and the second polynomial as a dividend, and proceed as before.

III. Continue the operation until a remainder is obtained which will exactly divide the preceding divisor; this last remainder, multiplied by the factor set aside, will be the greatest common divisor sought; if no remainder is found which will exactly divide the preceding divisor, then the factor set aside is the greatest common divisor sought.

We begin by dividing the polynomial of the highest degree by that of the lowest; the quotient is, as we see in the above table, a + 4b, and the remainder 19ab2 1963.

But,

19ab2 · 1963 = 1962 (a - b).

Now, the factor 1962, will divide this remainder without dividing

hence, the factor must be suppressed, and the question is reduced to finding the greatest common divisor between

a2-5ab4b2 and a - b.

Dividing the first of these two polynomials by the second, there is an exact quotient, a -4b; hence, a b is the greatest common divisor of the two given polynomials. To verify this, let each be divided by a - b.

2. Find the greatest common divisor of the polynomials,

3a5 - 5a3b2+2ab4 and

2a - 3a2b2 + b1.

We first suppress a, which is a factor of each term of the first polynomial: we then have,

3a4. 5a2b2 + 2b1 || 2a1 — 3a2b2 + ba.

We now find that the first term of the dividend will not contain the first term of the divisor. We therefore multiply the dividend by 2, which merely introduces into the dividend a factor not common to the divisor, and hence does not affect the common divisor sought. We then have,

We find after division, the remainder - a2b2 + b4 which we put under the form b2 (a2 - b2).

-

b2 (a2 — b2).

We then suppress