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in which the coefficients are all integers, and that of the first term unity.
The substitution of for x is not always the one which leads to the most simple result; but when A contains two or more equal factors, each factor need scarcely ever be repeated more than once.
3x2 5x 2 Ex. 1. Transform the equation 23 +
0 into an
2 4 other whose coefficients are integers, and that of the first term unity. Clearing of fractions, we have
36x3-54x2+4ox-8=0. Substituting for æ, the transformed equation is
y gy2 45y
8 = 0,
73-9y2 +457–48=0. Transform the following equations into others whose coefficients are integers, and that of the first term unity.
1 Ex. 2. 2+2x2 + -0. Ans. yu+1242 +9y-24=0.
4 9 Ex. 3. &c*+ +2=0. Ans. 73+2y2-9y+432=0. Ex. 4. **+2+x2 +*3=0. Ex. 5. * -44+8x+21=0.
14x2 10 Ex. 6. 23.
442. If in any complete equation involving but one unknown quantity the signs of the alternate terms be changed, the signs of all the roots will be changed. Take the general equation of the nth degree,
2n + A.xn-1 + BaC"-9 + Can-3+ ... =0. (1.) in which the signs may follow each other in any order whatever.
If we change the signs of the alternate terms, we shall bave
" -- Axh-1+ Bxn-2-CX-3+.... =0. (2.) or, changing the sign of every term of the last equation,
-X" + A2-1-B.xn-2+ C.*-3-.... =0. (3.) Now, substituting ta for a in equation (1) will give the same result as substituting -a in equation (2), if n be an even number; or substituting -a in equation (3), if 'n be an odd number. If, then, a is a root of equation (1), - a will be a root of equation (2), and, of course, a root of equation (3), which is identical with it.
Hence we see that the positive roots may be changed into negative roots, and the reverse, by simply changing the signs of the alternate terms; so that the finding the real roots of any equation is reduced to finding positive roots only.
This rule assumes that the proposed equation is complete ; that is, that it bas all the terms which can occur in an equation of its degree. If the equation be incomplete, we must introduce any missing term with zero for its coefficient.
Ex. 1. The roots of the equation 23–202 –5x+6=0 are 1, 3, and —2; what are the roots of the equation
23 +2.2 - 5x— 6=0? Ex. 2. The roots of the equation 23— 6x2 +113—6=0 are 1, 2, and 3; what are the roots of the equation
23 + 6x2 +11+6=0? Ex. 3. The roots of the equation ** — 6x3 +5x2+2x–10=0 are -1, +5,1+V-1, and 1-V-1; what are the roots of the equation
* +603 +522-23-10=0?
443. If an equation whose coefficients are all real contains imaginary roots, the number of these roots must be even.
If an equation whose coefficients are all real has a root of the form a+b7—1, then will a-bv-1 be also a root of the equation. For, let a+bV-1 be substituted for x in the equation, the result will consist of a series of terms, of which those involving only the powers of a and the even powers of bv – 1 will be real, and those which involve the odd powers of 6V - 1 will be imaginary.
If we denote the sum of the real terms by P, and the sum of the imaginary terms by QV-1, the equation becomes
P+QV-1--0. But, according to Art. 243, this equation can only be true when we have separately P=0 and Q=0.
If we substitute a-bv-1 for x in the proposed equation, the result will differ from the preceding only in the signs of the odd powers of bV-1, so that the result will be P-QV-1. But we have found that P=0 and Q=0; hence P-QV-1=0. Therefore a-bv-1, when substituted for x, satisfies the equation, and, consequently, it is a root of the equation.
It may be proved in a similar manner that if an equation whose coefficients are all rational, has a root of the form a+V6, then will a-Vibe also a root of the equation.
Ex. 1. One root of the equation 23—2x+4=0 is 1+V-1; what are the other roots ?
Ex. 2. One root of the equation 23 - 2 - 7x + 15 =0 is 2+V-1; what are the other roots ?
Ex. 3. One root of the equation 23 — 202 + 3x +5=0 is 1+2V-1; what are the other roots ?
Ex. 4. One root of the equation 2* — 4x3 + 4x – 1=0 is 2+V3; what are the other roots ? Ex. 5. Two roots of the equation
28 +236 +4x5 +4x4 -882-162—32=0 are -1+1-1 and 1-V3; what are the other six roots ?
444. Any equation involving but one unknown quantity may
be transformed into another whose roots differ from those of the proposed equation by any given quantity.
Let it be required to transform the general equation of the nth degree into another whose roots shall be less than those of the proposed equation by a constant difference h.
Assume y=x-h, whence x=y+h.
(y+h)" +A(y+h)"-1+By+h)-2+.... +V=0.
Developing the different powers of yth by the binomial formula, and arranging according to the powers of y, we have g+ nh|y--++n(n-1)h2|yo- +n(n-1)(n—2)23]y*-* +, etc. +A
+C which equation satisfies the proposed condition, since y is less than ~ by h. If we assume y=x+h, or x=y-h, we shall obtain in the same manner an equation whose roots are greater than those of the given equation by h.
Ex. 1. Find the equation whose roots are greater by 1 than those of the equation 3 + 3x2 - 4x+1=0. We must here substitute y-1 in place of x.
Ans. yo—77+7=0. Ex. 2. Find the equation whose roots are less by 1 than those of the equation 23-2x +32-4=0.
Ans. yo + y2 +2y—2=0. Ex. 3. Find the equation whose roots are greater by 3 than those of the equation 2+ +933 +122–14x=0.
Ans. Y4 — 343–1542 +497–12=0. Ex. 4. Find the equation whose roots are less by 2 than those of the equation 53* — 12x + 3x2 + 4x–5=0.
Ans. 544 +2843 +51y2 +327-1=0. Ex. 5. Find the equation whose roots are greater by 2 than those of the equation +10x* +42% +86x2 +70x+12=0.
Ans. yö +2y3 - 6y2-10y+8=0. 445. Any complete equation may be transformed into another whose second term is wanting.
Since h in the preceding article may be assumed of any value, we may put nh+A=0, which will cause the second term
of the general development to disappear. Hence h=-A and
Hence, to transform an equation into another which wants the second term, substitute for the unknown quantity a new unknown quantity minus the coefficient of the second term divided by the highest exponent of the unknown quantity.
Ex. 1. Transform the equation <3 — 6x2 + 8x—2=0 into an other whose second term is wanting. Put x=y+2.
Ans. 43-47-2=0. Ex. 2. Transform the equation 24 — 16x3— 6x+15=0 into another whose second term is wanting. Put x=y+4.
Ans. y - 96y2-518y-777=0. Ex. 3. Transform the equation
25 + 15* + 12x3 20x2 +14_-25=0 into another whose second term is wanting.
Ans. yo—7843+412y — 757y+401=0. Ex. 4. Transform the equation X4 - 8x3+5=0 into another whose second term is wanting.
According to Art. 438, when the second term of an equation is wanting, the sum of the positive roots is numerically equal to the sum of the negative roots.
446. If two numbers, substituted for the unknown quantity in an equation, give results with contrary signs, there must be at least one real root included between those numbers.
Let us denote the real roots of the general equation of the nth degree by a, b, c, etc., and suppose them arranged in the order of their magnitude, a being algebraically the smallest, that is, nearest to - ; b the next smallest, and so on. The equation may be written under the following form,
(x-a)(x-7)(x-c)(x-d)..... =0. Now let us suppose x to increase from
toward ta, assuming, in succession, every possible value. As long as x is less than a, every factor of the above expression will be negative, and the entire product will be positive or negative according as the number of factors is even or odd. When a becomes equal to a, the whole product becomes equal to 0. But if x be greater than a and less than 6, the factor x—a will be positive, while all the other factors will be negative. Hence, when changes from a value less than a to a value greater than a and less than b, the sign of the whole product changes from + to or from – to +. When a becomes equal to b, the product agai: becomes zero; and as x increases from b to c, the factor