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Every equation of the mth degree has m roots, and can have no

more.

282. In equations which arise from the multiplication of equal factors, such as

(x − a)1 (x —b)3 (x — c)2 (x — d) = 0,

the number of roots is apparently less than the number of units in the exponent which denotes the degree of the equation. But this is not really so; for, the above equation actually has ten roots, four of which are equal to a, three to b, two to c, and one to d.

It is evident that no quantity a', different from a, b, c, d, can verify the equation; for, if it had a root a', the first member would be divisible by xa', which is impossible.

Consequence of the second Property.

283. It has been shown that the first member of every equation of the mth degree, has m binomial divisors of the first degree, of the form

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If we multiply these divisors together, two and two, three and three, &c., we shall obtain as many divisors of the second, third, &c. degree, with reference to r, as we can form different combinations of m quantities, taken two and two, three and three, &c. Now the number of these combinations is expressed by

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Composition of Equations.

284. If in the identical equation

xm + Pxm¬1 + . . . = (x − a) (x − b) (x − e). . . (x − 1),

...

we perform the multiplication of four factors in the second member, we have,

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If we perform the multiplication of the m factors of the second member, and compare the terms of the two members, we shall find the following relations between the co-efficients P, Q, R, . . . T, U, and the roots a, b, c,... k, l, of the proposed equation, viz.,

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The double sign has been placed in the last relation, because the product ax-bx -c.. × I will be plus or minus according as the degree of the equation is even or odd. Hence,

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1st. The algebraic sum of the roots, taken with contrary signs, is equal to the co-efficient of the second term; or, the algebraic sum of the roots themselves, is equal to the co-efficient of the second term taken with a contrary sign.

2d. The sum of the products of the roots taken two and two, with their respective signs, is equal to the co-efficient of the third

term.

3d. The sum of the products of the roots taken three and three, with their signs changed, is equal to the co-efficient of the fourth term; or the co-efficient of the fourth term, taken with a contrary sign, is equal to the sum of the products of the roots taken three and three; and so on.

4th. The product of all the roots, is equal to the last term; that is, the product of all the roots, taken with their respective signs, is equal to the last term of the equation, taken with its sign, when the equation is of an even degree, and with a contrary sign, when the equation is of an odd degree. If one of the roots is equal to 0, the absolute term will be 0.

The properties demonstrated (Art. 143), with respect to equations of the second degree, are only particular cases of the above.

Consequences.

1. If the co-efficient of the second term of an equation is equal to zero, the term will not appear in the equation; and the sum of the positive roots is equal to the sum of the negative roots. 2. Every commensurable root of an equation is a divisor of the last or absolute term.

EXAMPLES IN THE FORMATION OF EQUATIONS.

6.

1. Form the equation whose roots are 2, 3, 5, and We have, by simply indicating the multiplication of the factors, (x-2)(x-3) (x — 5) (x+6)= 0.

But the process may be shortened by detaching the co-efficients thus:

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

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3. What is the equation whose roots are 3,

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7x+6=0.

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

and - 6?

Ans. x3

32x + 24 =

0.

5. What is the equation whose roots are 1,

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Of the greatest Common Divisor.

285. The greatest common divisor of two polynomials is the greatest polynomial, with reference to its exponents and co-efficients, that will exactly divide the proposed polynomials.

If two polynomials be divided by their greatest common divisor, the quotients will be prime with respect to each other; that is, they will no longer contain a common factor. Hence,

Two polynomials are prime with respect to each other when they have not a common factor.

Let A and B be two polynomials, D their greatest common divisor, and A', B', the quotients after division. Then

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Now, if A' and B' have a common factor d, then d x D would be a common divisor of the two polynomials and greater than D, either with respect to the exponents or the co-efficients, which would be contrary to the supposition.

Again, since D exactly divides A and B, every factor of D will have a corresponding factor in both A and B. Hence,

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

286. We will now show that the greatest common divisor of two polynomials will divide their remainder after they have been divided by each other.

Let A and B be two polynomials, D their greatest common divisor, and suppose A to contain the highest exponent of the letter with reference to which the polynomials A and B are arranged.

Having divided A by B, suppose we have a quotient Q and a remainder R. We may then write

A =BXQ + R.

If now, we divide both members of the equation by D, we have

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and since we suppose A to be divisible by D, the first member of the equation will be entire, and consequently, the second member must also be entire, since an entire quantity cannot be equal to a fraction. But since D also divides B, the first term of the second member is entire, and consequently, the second term is also entire, and therefore, R is exactly divisible by D.

We will now show that if D will exactly divide B and R, that it will also divide A. For, having divided A by B, as before, we have

A = B × Q + R, and by dividing by D, we obtain

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But since we suppose B and R to be divisible by D, and know Q to be an entire quantity, the second member of the equation is entire hence, the first member is also entire, that is, A is exactly divisible by D. Hence,

2dly. The greatest common divisor of two polynomials, is the same as that which exists between the least polynomial and their remainder after division.

REMARK.-If either of the polynomials A or B have a factor A' common to all its terms, but not common to the other polynomial, the common divisor will be found in that part of the polynomial which is multiplied by the factor A'.

287. From these principles, we have, for finding the greatest common divisor of two polynomials, the following

RULE.

I. Take the first polynomial and suppress all the monomial factors common to each of its terms. Do the same with the second polynomial, and if the factors so suppressed have a common divisor, set it aside as forming a part of the common divisor sought.

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