these is b (2) Form all possible products of r 1 of the negab. tive roots excepting Bm+1, and multiply each product by -Pm+1 Add all the products obtained. This process, it is observed, is precisely that indicated by (3). REMARK. It is noticed that in the rule the signs of the roots are always changed before forming any term. This does not involve any change when r is an even number, but is included in the rule for the sake of uniformity. COROLLARY. Every root of an equation is a factor of its constant term. 171. The general term in the binomial expansion. On p. 129 we gave an expression for the rth term of the binomial expansion, the validity of which we can now establish. In (1), § 170, let B1 = B2 = · B. Denote this common value by a. The expression (1) becomes, on writing n in place of m, == By the theorem in § 170, b, is the sum of all possible products of r of the negative roots. Since there are as the form of the rth term of the expansion of (x + a)". 172. Solution by trial. Since by the previous corollary every root of an equation is a factor of its constant term, we may in many cases test by synthetic division whether or not a given equation has integral roots. Thus the integral roots of the equation x1 – 8 x3 + 4 x2 + 24 x − 21 = 0 must be factors of 21. We try + 1 by synthetic division, 1-8+4+24-211 +1-73 +21 1-7-3+21 0 (1) Thus 1 is a root of (1), and the quotient of the equation by If this equation has any integral root it must be a factor of 21. We try + 3 by synthetic division, Thus 7 is a root, and the remaining roots of (1) are the roots of Hence the roots of (1) are +1, + 7, ± √3. 173. Properties of binomial surds. A binomial surd is a number of the form a ± √, where a and b are rational numbers, and where is positive but not a perfect square. Though we have not explicitly defined what we mean by the sum of an irrational number and a rational number, we shall assume that we can operate with the binomial surd just as we would be able to operate if b were a perfect square. THEOREM I. If a binomial surd a + √b=0, then a =0 and b = 0. If a + √b = 0 and either a = 0 or = 0, clearly both must equal zero. Suppose, however, that neither a nor b equals zero. Then transposing we have a = - √, and a rational number would be equal to an irrational number, which cannot be. Hence the only alternative is that both a and b equal zero. THEOREM II. If two binomial surds, as a + √b and c + √d, are equal, then a = c and b = d. Thus, by Theorem I, either b = 0, which is contrary to the definition of a binomial surd, or a — c = 0, that is, a = c. In the latter case (1) reduces to √ √d, or b = d, and we have a = c and b = d, which was to be proved. a + √b and a are called conjugate binomial surds. THEOREM III. If a given binomial surd a + √b is the root of an equation with rational coefficients, then its conjugate is also a root of the same equation. The proof of this theorem, which should be performed in writing by each student, may be made analogously to the proof of the theorem on p. 174. 174. Formation of equations. If we know all the roots of an equation, we may form the equation in either one of two ways (see p. 167 and p. 177). a FIRST METHOD. If α1, α, α, are the given roots, multiply together the factors xα, X — α „ SECOND METHOD. From the given roots form the coefficients by the rule on p. 177. If the equation and all but one of its roots are known, that root can be found by the solution of a linear equation obtained from the coefficient of the second or the last term. If all but two of its roots are known, the unknown roots may be found by the solution of a pair of simultaneous equations formed from the same coefficients. In the solution of the following exercises use is made of the theorem on p. 174, Theorem III, p. 180, and the various relations between the roots and the coefficients. EXERCISES 1. Form the equations which have the following roots. Check the process by using both methods of § 174. 2. The equation x + 2 x3-7 x2 - 8x + 12 = 0 has two roots - 3 and +1. Find the remaining roots. Solution: Let the unknown roots be a and b. 4. x3 Find the remaining roots. Find the remaining roots. 6. Two roots of x4 - 35 x2 + 90 x − 56 = 0 are 1 and 2. Find the remaining roots. 8. The two equations x3 −6 x2+11x−6 = 0 and x3 − 14 x2 + 63 x − 90 = 0 have a root common. Plot both equations on the same axes, and find all the roots of both equations. 9. Determine the middle term of the equation whose roots are - 2, + 1, 3, − 4 without determining any other term. 10. What is the last term of the equation whose roots are 11. One root of x4 - 4x3 + 5x2 + 2x + 52 = 0 is 3 - 2i. ing roots. - Find the others. 12. One root of 24. - 4x3 + 5 x2 + 8x-14=0 is 2+ i√3. 13. Plot the following equations, determine all the integral roots, and find the remaining roots by solving. (a) x4 6x3+24x 16 0. ΤΑ = In this plot two squares on the X axis represent a unit of x, while one square on the Y axis represents ten units of y. The integral factors are 2 and x+2, since 2 are roots, that is, are values of x for which the |