and we have Examples of finding Equal Roots. x3 — 7 x2 + 16 x-12= (x-2)2 (x —3) = 0; whence x=3 is the other root of the given equation. 2. Find all the roots of the equation x2 x79x56x4 + 15 x3- 12 x2 — 7x+6=0 which has equal roots. Solution. The derivative of this equation is 7x6 7, - 45 x2+24 x3 + 45 x2. -24 x the greatest common divisor of which and the given equation gives x3 — x2-x+1=0, which is an equation of the third degree, and we may consider it as a new equation, the equal roots of which are to be found, if it has any. Now its derivative is 3x22x-1, and the common divisor of this derivative and the first member gives ac -10, or x = 1. x3 — x2−x+1 = (x − 1)2 (x + 1) = 0. The equal roots of the given equation are, therefore, Examples of finding Equal Roots. and is found by division to be (x − 1)3 (x + 1)2 (x2 + x -6). The remaining roots are, therefore, found from solving the 3. Find all the roots of the equation x3-3x2-9x27 0 +3 which has equal roots. Ans. x 3, or —— 3. 4. Find all the roots of the equation 23-15x275x1250 which has equal roots. Ans. x = 5. 5. Find all the roots of the equation x4-9x329 x2-39x+18=0. which has equal roots. Ans. x= = 1, or = 2, or = 3. 6. Find all the roots of the equation x4 - 2 x3 - 59 x2 +60 x + 900 = 0 Number of Real Roots. 9. Find all the roots of the equation x5 — 2 x1 — 2 x3 +4x2+x-2=0 -- - · which has equal roots. 10. Find all the roots of the equation x6 6x44x3 +9 x2 - 12x+4=0 which has equal roots. Ans. x= 1, or = - 2. 11. Find the equal roots of the equation x8-8x+26x6-45x545x4-21 x3-10x2+20x-8=0. Ans. x = 1, or = 2. SECTION III. 281. Theorem. Real Roots. When an equation is reduced, as in art. 266, and the values of its first member, obtained by the substitution of two different numbers for its unknown quantity, are affected by contrary signs, the given equation must have a real root comprehended between these two numbers. Proof. For, if the value of the less of the two numbers, which are substituted for the unknown quantity is supposed to be increased by imperceptible degrees until it attains the value of the greater number, the value of the first member must likewise change by imperceptible degrees, and must pass through all the intermediate values between its two extreme values. But the extreme values are affected with opposite signs, so that zero must be contained between them, and must be one of the values attained by the first member; Number of Real Roots between two given Numbers. that is, there must be a number which, substituted in the first member, reduces it to zero, and this number is consequently a root of the given equation. 282. Corollary. If the given equation has no real root, the value of its first member will always be affected by the same sign, whatever numbers be substituted for its unknown quantity. 283. Theorem. When an uneven number of the real roots of an equation are comprehended between two numbers, the values of its first member obtained, by substituting these numbers for x, must be affected with contrary signs; but if an even number of roots is contained between them, the values obtained from this substitution must be affected with the same sign. Proof. Denote by x', x, x &c. all the roots of the given equation which are contained between the given numbers p and q; the first member of the given equation must, by art. 269, be divisible by (x — x') (x — x'') (x — x''') &c. If we denote the quotient of this division by Y, the equation Y = 0 gives all the remaining roots of the given equation, so that Y =0 cannot have any real root contained between P and q. The given first member being, therefore, represented by becomes (x — x') (x — x11) (x — x111) · .... X Y (p—x') (p—x'') (p—x'""') . . . . × Y', when we substitute p for x, and denote the corresponding Number of Real Roots between two given Numbers. value of Y by Y'; and when we substitute q for X, and denote the corresponding value of Y by Y", it becomes Now since each of the roots x', x'', x'", &c., is included between p and q, the numerator and denominator of each of the fractions must be affected with contrary signs, and therefore each of these fractions must be negative. But since Y and Y" must, by art. 282, have the same sign, the fraction is positive. Y' The product of all these fractions is therefore positive, when the number of the fractions is even, that is, when the number of the roots, x', x'', x''', &c., is even; and this product is negative, when the number of these roots is uneven. The values which the given first member obtains by the substitution of P and 9 for x must, consequently, be affected with contrary signs in the latter case; and with the same sign in the former case. |