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2C-6 becomes positive, and the sign of the product changes again from – to + or from + to ; and, in general, the product changes its sign as often as the value of x passes over a real root of the equation.
Hence, if two numbers substituted for in an equation give results with contrary signs, there must be some intermediate number which reduces the first member to 0, and this number is a root of the equation.
If the two numbers which give results with contrary signs differ from each other only by unity, it is plain that we have found the integral part of a root.
If two numbers, substituted for x in an equation, give results with like signs, then between these numbers there will either be no root, or some even number of roots. The last case may include imaginary roots.
For if a +bV-1 be a root of the equation, then will a-6V-1 be also a root. Now
(oc-a-bv-1)(x—a+bV-1)=(x—a)2 +62, a result which is always positive; that is, the quadratic factor corresponding to a pair of imaginary roots of an equation whose coefficients are real, is always positive.
Ex. 1. Find the first figure of one of the roots of the equation 23 + x2+x-100=0.
When x=4, the first member of the equation reduces to -16; and when x=5, it reduces to +55. Hence there must be a root between 4 and 5; that is, 4 is the first figure of one of the roots.
Ex. 2. Find the first figure of one of the roots of the equation 3 - 6x2 + 9x-10=0.
Ex. 3. Find the first figure of each of the roots of the equation 23-4x2 - 6x+8=0.
447. In a series of terms, two successive signs constitute a permanence when the signs are alike, and a variation when they are unlike. Thus, in the equation 2 – 2x2–5x+6=0, the signs of the first two terms constitute a variation, the signs of the second and third constitute a permanence, and those of the third and fourth also a variation.
Descartes's Rule of Signs. 448. Every equation must have as many variations of sign as it has positive roots, and as many permanences of sign as it has negative roots.
According to Art. 436, the first member of the general equation of the nth degree may be regarded as the product of n binomial factors of the form x-a, x-b, etc. The above theorem will then be demonstrated if we prove that the multiplication of a polynomial by a new factor, x—a, corresponding to a positive root, will introduce at least one variation, and that the multiplication by a factor, x+a, will introduce at least one perma
Suppose, for example, that the signs of the terms in the original polynomial are ++---+-+--t, and we have to multiply the polynomial by a binomial in which the signs of the terms are +-. If we write down simply the signs which occur in the process and in the result, we have
+++ t-It We perceive that the signs in the upper line of the partial products must all be the same as in the given polynomial; but those in the lower line are all contrary to those of the given polynomial, and advanced one term toward the right. When the corresponding terms of the two partial products have different signs, the sign of that term in the result will depend upon the relative magnitude of the two terms, and may be either + or Such terms have been indicated by the doublo sign #; and it will be observed that the permanences in the given polynomial are changed into signs of ambiguity. Hence, take, the ambiguous sign as you will, the permanences in the final product are not increased by the introduction of the positive root +a, but the number of signs is increased by one, and
therefore the number of variations must be increased by one. Hence each factor corresponding to a positive root must introduce at least one new variation, so that there must be as many variations as there are positive roots.
In the same manner we may prove that the multiplication by a factor, w+a, corresponding to a negative root, must intro. duce at least one new permanence; so that there must be as many permanences as there are negative roots.
If all the roots of an equation are real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. If the equation is incomplete, we must supply the place of any deficient term with #0 before applying the preceding rule.
Ex. 1. The equation 2 — 3x4 — 5x3+15x2 +4x—12=0 ha five real roots; how many of them are positive ?
Ex. 2. The equation 24-3x3-15x2 +49x-12= has four real roots; how many of them are negative? Ex. 3. The equation
2 + 3x5 –4124 -8733 +400x2 +444x— 720=0 has six real roots; how many of them are positive?
Derived Polynomials. 449. If we take the general equation of the nth degree, and substitute y+h in place of x, it becomes
(y+h)" +A(y+1)n-1+B(y+h)n-2+.... +V=0. Developing the powers of the binomial y+h, and arranging in the order of the powers of h, we have
h2 y" +nyn-1 h+n(n-1)yn-3 + Ayn-1+(n-1)Ayn-3 +(n-1)(n-2)Ayn-3 +Byn-2 + (n − 2)Byn-3 +(n—2)(n-3)Byn-4
+T + V
The part of this development which is independent of h is of the same form as the original polynomial, and we will des.
ignate it by X. We will denote the coefficient of h by Xy, the coefficient of
by X,, etc. The preceding development may then be written
450. The polynomials X, X,, etc., are called derived polynomials, or simply derivatives. X, is called the first derivative of X, X, the second derivative, and so on. X is called the primitive polynomial. Each derived polynomial is deduced from the preceding by multiplying each term by the exponent of the leading letter in that term, and then diminishing the exponent of the leading letter by unity. Ex. 1. What are the successive derivatives of 23 -70+8x-3?
1st. 3.2 - 14x+8. Ans. 2d. 6x-14.
X* - 8x3+14x2+4x-8?
25+ 3x4 + 2003 - 3x2 - 2x2?
X" + An-1 + Ban-2+.... + Tx+V?
Equal Roots. 451. We have seen, Art. 436, that if a, b, c, etc., are the roots of the general equation of the nth degree, the equation may be written
X=(x-a)(2–0).—c).... (x-k)(x-1)=0. | When the equation has two roots equal to a, there will be two factors equal to x-a; that is, the first member will be divisible by (2-a)?; when there are three roots equal to a, the first member will be divisible by (x— ); and if there are n roots equal to a, the first member will contain the factor (-a)”. The first derivative will contain the factor n(x-a)n-1; that is,
-a occurs (n-1) times as a factor in the first derivative. The greatest common divisor of the primitive polynomial, and its first derivative, must therefore contain the factor -a, repeated once less than in the primitive polynomial.
Hence, to determine whether an equation has equal roots, we have the following
RULE. Find the greatest common divisor between the given polynomial and its first derivative. If there is no common divisor the equation has no equal roots. If there is a common divisor, place this equal to zero, and solve the resulting equation. Ex. 1. Find the equal roots of the equation
23 – 8x2 +21x-18=0. The first derivative is 3.2 – 16x+21.
The greatest common divisor between this and the given polynomial is -3.
IIence the equation has two roots, each equal to 3.
2 – 13x2 +55% -75=0.
Ans. Two roots equal to 5. Ex. 3. Find the equal roots of the equation
23 — 7x2 +16x—12=0.
Ans. Two roots equal to 2. Ex. 4. Find the equal roots of the equation
24 - 6x2-8x_3=0..
Ans. Three roots equal to – 1. Ex. 5. Find the equal roots of the equation
23-302-9x+27=0. Ex. 6. Find the equal roots of the equation
23 + 8x2 + 20x+16=0.
Sturm's Theorem. 452. The object of Sturm's theorem is to determine the number of the real roots of an equation, and likewise the situation of these roots, or their initial figures when the roots are irrational.
According to Art. 446, if we suppose x to assume in succession every possible value from – o to + a, and determine the