2. Let the roots of x-x2+x-7=0, be squared. Whence y−y+ √y—7=0, the equation required. OF THE LIMITS OF THE ROOTS OF 42. Let x-a.x-b.x—c.x+d=o, be an equation, having the root a greater than b, b than c, and c than d"; "in which, if a quantity greater than a be substituted for x, (as every factor is, on this supposition, positive,) the result will be positive; if a quantity less than a, but greater than b, be substituted, the result will be negative, because the first factor will be negative, and the rest positive. If a quantity between b and c be substituted, the result will again be positive, because the two first factors are negative, and the rest positive; and so on °. Thus, transformed equation are 2.56, and 112.36, which are the squares of the former respectively. "In this series the greater is d, the less is -d; and whenever a, b, c,—d, &c. are said to be the roots of an equation, taken in order, a is supposed to be the greatest. Also in speaking of the limits of the roots of an equation, we understand the limits of the possible roots." This note, and the article to which it refers, were taken from Mr. Wood's Algebra; see likewise, on this subject, Maclaurin's Algebra, part 2. ch. 5. Wolfius's Algebra, part 1. sect. 2. ch. 5. Sir Isaac Newton's Arithmetica Universalis, p. 258. &c. Dr. Waring's Meditationes Algebraicæ, &c. • To illustrate this, let the roots of the equation x —px3 +qx2 —rx+s=0 be a, b, c, and d; then x-a-o, x—b=o, x—co, and x-d=0; and let g, which we will suppose less than a, but greater than b, be substituted for x in the latter equations; then will g-a be negative, and the rest, viz. g-b, g-c, and g―d, positive, and consequently their product will be positive; and g—a, (a negative quantity,) multiplied into this positive result, will therefore give a negative product: if h, which is less than b, but greater than c, be substituted for x, we have h-a and h-b both negative, and their product positive; but he and h-d are both negative, therefore their product is posi quantities which are limits to the roots of an equation, (or between which the roots lie,) if substituted for the unknown quantity, give results alternately positive and negative." 43. " : Conversely, if two magnitudes, when substituted for the unknown quantity, give results one positive and the other negative, an odd number of roots must lie between these magnitudes and if as many quantities be found as the equation has dimensions, which give results alternately positive and negative, an odd number of roots will lie between each two succeeding quantities; and it is plain that this odd number cannot exceed unity, since there are no more limiting terms than the equation has dimensions." 44. If when two magnitudes are severally substituted for the unknown quantity, both results have the same sign, either an even number of roots, or no root, lies between the assumed magnitudes. COR. Hence, any magnitude is greater than the greatest root of the equation, which, being substituted for the unknown quantity, gives a positive result. 45. To find a limit greater than the greatest root of an equation. RULE. Diminish the roots of this equation by the quantity e, (Art. 36.) and if such a value of e can be found, as shall make every term of the transformed equation positive, all its roots will be negative, (Art. 31. Cor.) consequently e will be greater than the greatest root of the equation. EXAMPLES.-1. To find a limit greater than the greatest root of x2-5x+6=0. Whence (y2+2 ye-5y+e2-5e+6=o, or) y2+2e-5.y +e.e−5+6=0, is the transformed equation; now it appears by trials, that 4 being substituted for e in this equation, it will be tive; and these two products multiplied, give likewise a positive product. In like manner it may be shewn, by substituting k, which is less than c, and greater than d, the result will be negative; and substituting m, less than the least root, the result will be positive. come y2+3y+2=0, of which all the roots are negative; wherefore 4 is greater than the greatest root of the equation x2−5x+ 6=0. 2. To find a limit greater than the greatest root of r3—12r2 +411-43=0. Wherefore (y+3y2e-12 y2+3 ye2-24ye+41 y+e—12e2+4le —43=0, or) y3 +3.e— 12.y2+3 e2−24e+41.y+e.e3 — 12 e+41 —43=0, is the transformed equation, where (by trials) it is found, that if 8 be substituted for e, the terms will be all positive; viz. y3 + 12 y3 +41 y+29=o; whence 8 is greater than the greatest root of the given equation. 3. Required a limit greater than the greatest root of r3—6x2 --25x- -12=o. Ans. 9. 4. Find a limit greater than the greatest root of xa—5x3+ 6x2−7x+8=0. 5. To find a limit greater than the greatest root of xa +3x3— 5r+82–20=0. 46. To find a limit less than the least root of an equation. RULE. Change the signs of the even terms, (the second, fourth, sixth, &c.) and proceed as before; then will the limit greater than the greatest root of the transformed equation, with its sign changed, be less than the least root of the given equation. See Cor. to Art. 30. and Art. 45. EXAMPLES.-1. Let x3-7x+8=0, be given to find a limit less than the least of its roots. This equation, by changing the sign of its second term, becomes x2+7x+8=0. Whence (y+2ye+7 y+e* +7e+8=0, or) y2+2e+7.y +e+7.e+8=0, is the transformed equation; and if-1 be substi tuted for e, all its terms will be positive, for the equation becomes y3 +5y+2=0; wherefore +1 is a limit less than the least root of the equation x2 -7x+8=0. 2. To find a limit less than the least root of x3+x2-10x+ 6=0. Changing the signs of the second and fourth terms, the equation becomes x3 — x2 - 10x-6=o. Whence y3+3e-1.y2+3 e2 —2e−10.y+e2 —e—10.e—6 =0, is the transformed equation, in which 4 being substituted for e, it becomes y3+11 y2+30y+2=0; wherefore —4 is less than the least root of the equation x3 + x2 — 10x+6=0. 3. To find a limit less than the least root of x2+12x-20 4. To find a limit less than the least root of x3-4x3-5x+ 6=0. 5. To find a limit less than the least root of x*—5 x3 —3—o. 6. To find the limits of the roots of r3+x2 — 10x+9=0. Ans.+3 and 5. 7. Required the limits of x-4x3+8x2 - 14x+20=o? 1 8. What are the limits of the roots of x3-2x2 -5x+7=0? 9. What are the limits of the roots of x1+3x2 −5x+10=o? RESOLUTION OF EQUATIONS OF 47. When the possible roots of an equation are integers, either positive or negative, they may be discovered as follows. RULE I. Find all the divisors of the last term, and substitute them successively for the unknown quantity in the proposed equation. II. When by the substitution of either of these divisors for the root, the resulting equation becomes=0, that divisor is a root of the given equation; otherwise it is not. III. If none of the divisors succeed, the roots are either fractional, irrational, or impossible. IV. When the last term admits of a great number of divisors, it will be convenient to transform the given equation into another, (Art. 35, 36.) the last term of which will have fewer divisors. 3 EXAMPLES.-1. Let x3-2x2 -5x+6=0, be given, to find its integral roots by this method. First, the divisors of the last term 6, are+1,−1, +2, −2, +3,−3, +6, and −6; now +1 being substituted for x in the given equation, it becomes +1−2−5+6=0; wherefore +1 is a root. Next, let - 1 be substituted, and the equation becomes —1—2 +5+6=8; wherefore -1 is not a root. Thirdly, let +2 be substituted, and the equation becomes 8—8—10+6=-4; wherefore+2 is not a root. Fourthly, let-2 be substituted, and the equation becomes —8—8+10+6=0; wherefore −2 is a root. Fifthly, let +3 be substituted, and the equation will then become+27-18-15+6=0; wherefore +3 is likewise a root. Thus, the three roots of the given equation are +1,−2, and +3; and it is plain there can be no more than three roots, since the equation arises no higher than the third degree; consequently there is no necessity to try the remaining divisors. 2. Given x — 6 x2. 16x+21=0, to find the roots. The divisors of the last term 21, are +1,−1, +3,−3, +7, -7,+21, and -21; these being successively substituted for x, we shall have +21 -21 +194481-2646–336+21=191520 +194481-2646+336+21=192192 Wherefore +1 and +3 are the only roots which can be found by this method; the two remaining roots are therefore impossible, being-2-3. 1 |