CHAPTER XIV. INEQUALITIES. 174. Different expressions containing any given letter will have their values changed when different values are assigned to that letter; one may be for some values of the letter greater than the other, and for some values of the letter smaller than the other. 175. Two expressions, however, may be so related that, whatever values may be given to the letter, one of the expressions cannot be greater than the other. Thus, 2xx2+1, whatever value be given to x. NOTE. The signs ← and ✈ are read not less than and not greater than, respectively. 176. For finding whether this relation holds between two expressions, the following is a fundamental proposition: If a and b are unequal, a2+b2>2ab. or For, (a - b) must be positive, whatever the values of a and b. That is, (a - b)2 > 0, a2 - 2ab+b2 > 0. .. a2 + b2 > 2 ab. 177. The principles applied to the solution of equations may be applied to inequalities, except that if each side of an equality have its sign changed, the inequality will be reversed. (1) If a and b are positive, show that a+b3>ab+ab2. We shall have if (dividing each side by a + b), But this is true (3 176). a3 + b3> a2b + ab2, a2 - ab + b2 > ab, a2 + b2 > 2 ab. (2) Show that a+b+c>ab + ac+bc. Show that, the letters being unequal and positive: 1. a2+362>2b (a+b). 3. (a2+b2)(a*+b1) > (a3 +b3)2. 2. a3b+ab3> 2a2b2. 4. a2+a2c+ab2 + b2c + ac2 + bc2 >6abc. 5. The sum of any fraction and its reciprocal > 2. 6. If x=a+b2, and y=c2+d, xy ac+bd, or ad+bc. 7. ab+ac+bc<(a+b−c)2+(a+c—b)2+(b+c−a)3. 8. Which is the greater, (a2 + b2)(c2 + d2) or (ac + bd)2? 9. Which is the greater, aa— ba or 4 a3 (a−b) when a>b? CHAPTER XV. INVOLUTION AND EVOLUTION. 178. Involution. The operation of raising an expression to any required power is called involution. Every case of involution is merely an example of multiplication, in which the factors are equal. am 179. Index Law. If m is a positive integer, by definition =αΧαΧα to m factors. Consequently, if m and n are both positive integers, (a”)m = a′′ × a′′ × an to m factors. == (αχα ..... § 19 to n factors)(a x a to n factors) taken m times 180. If the exponent of the required power is a composite number, the exponent may be resolved into prime factors, the power denoted by one of these factors found, and the result raised to a power denoted by another factor of the exponent; and so on. Thus, the fourth power may be obtained by taking the second power of the second power; the sixth by taking the second power of the third power. 181. From the Law of Signs in multiplication it is evident that all even powers of a number are positive; all odd powers of a number have the same sign as the number itself. Hence, no even power of any number can be negative; and the even powers of two compound expressions which have the same terms with opposite signs are identical. Thus, (b − a)2 = {(a — b)}2 = (a — b)2. 182. Binomials. By actual multiplication we obtain,. (a+b)2= a2+2ab+b2; (a+b)3 a3 +3a2b+3ab2 + b3 ; = (a + b)* = a*+4a3b+6a2b2+4ab3 +ba. In these results it will be observed that: I. The number of terms is greater by one than the exponent of the power to which the binomial is raised. II. In the first term, the exponent of a is the same as the exponent of the power to which the binomial is raised; and it decreases by one in each succeeding term. III. b appears in the second term with 1 for an exponent, and its exponent increases by 1 in each succeeding term. IV. The coefficient of the first term is 1. V. The coefficient of the second term is the same as the exponent of the power to which the binomial is raised. VI. The coefficient of each succeeding term is found from the next preceding term by multiplying the coefficient of that term by the exponent of a, and dividing the product by a number greater by one than the exponent of b. If b is negative, the terms in which the odd powers of b occur are negative. Thus, (a — b)* = a* — 4 a3b+6a2b2 -- 4ab3 + ba. By the above rules any power of a binomial of the form a+b, or ab, may be written at once. 183. The same method may be employed when the terms of a binomial have coefficients or exponents. (1) (a — b)3 – a3 — 3 a2b + 3 ab2 — b3. = (2) (5 x2-2y3)3, = = (5 x2)3 — 3 (5 x2)2(2 y3) + 3 (5 x2)(2 y3)2 — (2 y3)3, = In like manner, a polynomial of three or more terms may be raised to any power by inclosing its terms in parentheses, so as to give the expression the form of a binomial. = = = (x3 − 2 x2)2 + 2 (x3 − 2 x2)(3 x + 4) + (3 x + 4)2, · x6 − 4x5 + 4x2 + 6 x1 — 4x3 — 16 x2 + 9 x2 + 24x + 16, 26-4x5+10x4 - 4x3- 7 x2 + 24 x + 16. Exercise 72. |