Hence the rationalizing factor of √5 √2 is 5§ +5121 +5821 + 5121 + 5121 + 2a. This factor reduces to 525 +55√√2+10+2√25√√2+4√5+4√2. — (ii.) If n be even, V+ Vy can be rationalized, since EXERCISE XX. 12 (This exercise may be omitted when the chapter is read for the first time.) Find the rationalizing factor of each of the following binomial surds: 48. Extending the idea of "factor" to include expressions in which surd numbers appear among the coefficients, we may, by applying the principles of Chapter XII., transform certain expressions so that they shall appear as products of factors involving surds. We will now consider the problem of factoring the general expression of the second degree containing one unknown, x: In the trinomial square a2± 2ab+b2 the third term 2 is the square of the quotient obtained by dividing the "middle term " or "finder term" ± 2 ab by twice the square root of the first term a2. (See Chapter XII. § 21.) That is, ± 2 ab\2 b = 2 a The process of obtaining a trinomial square, a2 ± 2 ab + b2, by adding the term b2 to a binomial such as a2± 2 ab, is called completing the square with reference to a2± 2 ab. Chapter XXII. §§ 18-20.) b (See also We may complete the square with reference to the binomial 2 + b C a [ a which appears in the expression a x2+x+ a a ] as follows: a by twice the square root of the which is the term whose square must be added to complete the square with reference to a2 + Using 6x as a "finder term," we may complete the square with reference The square may be completed with reference to 9 x2 + 24x by using 24 x as a "finder terin," as follows: We have, Hence, 3x2+8x-5=}[9x2 + 24x + 42 16 15] = } [(3x+4)2 - (√31)2] = [3x+4+√31][3x+4-√31]. Observe that in each of the examples above the factors obtained are of the first degree with reference to the letters appearing in them. EXERCISE XX. 13' Obtain factors containing surds for each of the following: 49. A root of a monomial surd may be found by applying the 50.* In the statements and proofs of the following principles, the radicands are restricted to positive commensurable values. (i.) The product or the quotient of two similar quadratic surds is rational. For if a, b, and c be rational numbers, a√c × b√c=ab√c2 = abc. (ii.) The product or the quotient of two dissimilar quadratic surds is a quadratic surd. For, in simplest form, every quadratic surd has as a radicand one or more prime factors raised to the first power only. Two dissimilar surds cannot have all of these factors alike, and accordingly their product must, after it is simplified, have at least one of these factors to the first degree as a radicand. From this it follows that (iii.) The sum or the difference of two dissimilar quadratic surds cannot be equal either to a rational number or to a single surd. * This section may be omitted when the chapter is read for the first time. Hence, we should have the product of two dissimilar quadratic surds equal to a rational number, which is impossible by (ii.) above. It follows that va± √√b cannot be equal to c, when a ‡ b. A similar method of proof holds for (2). (iv.) The square root of a rational number cannot be expressed as the sum of another quadratic surd and a rational number. That is, if a and b are quadratic surds, and c is any rational number, it follows that √a + √b + c. That is, if (1) be true, we have in (2) a surd number √ equal to a rational expression, which is impossible. Accordingly, a cannot be equal to √b+c. (v.) In any equation containing quadratic surds and rational numbers, the surd numbers in one member are equal to the surd numbers in the other member, and the rational numbers in one member are equal to the rational numbers in the other member. That is, if + y = √a + b, x (1) it follows that = a, and y = b, where a, b, x, and y are all commensurable numbers and Va and √ are surds. For, if yb, let y = b±n, where n 0. (2) |