Examples of finding Equal Roots. the first member, each being repeated once less than in the first member. No one of them is, then, a factor of the common divisor, unless it is more than once a factor of the first member, that is, unless it corresponds to one of the equal roots. The equal roots of an equation are, therefore, obtained by finding the greatest common divisor of its first member and its derivative, and solving the equation obtained from putting this common divisor equal to zero. 279. Corollary. The common divisor must, itself, have equal roots, whenever a root is more than twice a root of the given equation. which has equal roots. Solution. The derivative of this equation is 3x2-14 x + 16, the greatest common divisor of which and the given first member is Now since the given equation has two roots equal to 2, it must be divisible by (x-2)2= x2 - 4 x +4, 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 x79x56x4 + 15 x3 — 12 x2 -7x+6=0 which has equal roots. Solution. The derivative of this equation is 7x6-45x424 x3 +45 x2-24 x — 7, 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 3x22x1, and the common divisor of this derivative and the first member gives х -- 1 = 0, 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 x = 2, or =- - 3. 3. Find all the roots of the equation 233x29x27 0 = which has equal roots. Ans. x 3, or=3. 4. Find all the roots of the equation which has equal roots. Ans. x5. 5. Find all the roots of the equation x-9x329x2-39x+18=0 7. Find all the roots of the equation x4 6 x2-8x-3=0 which has equal roots. Ans. x=3, or —— =-1. 8. Find all the roots of the equation x2 + 12 x3+54x2 + 108x+81 = 0 which has equal roots. Ans. x 3. Number of Real Roots. 9. Find all the roots of the equation x52x42x3 + 4x2+x-2=0 which has equal roots. Ans. x=1, or = 2. 10. Find all the roots of the equation 26-6x+4x+9 x2-12x+4=0 which has equal roots. Ans. x = 1, or =— 2. 11. Find the equal roots of the equation 8-8x+26x6-45x5+45x4-21 x3-10x2+20x-8=0. Ans, x = 1, or = 2. SECTION ш. 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 Р and q; the first member of the given equation must, by art. 269, be divisible by (x — x') (x — x1'') (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 Р and 4. The given first member being, therefore, represented by becomes (x — x') (x — x'') (x — x'') . . . . × Y (p-x') (p-x)(p-x"").... x Y', when we substitute p for x, and denote the corresponding |