33. by 34. by 28. x2xy + y2 + x + y + 1 by x + y - 1. 29. 3 ab+2 a2b3 - 41 a3b2 by 24 a3 — 2} a2 + ab3. an-3xs+an by a2x2-1 — 3x2+1 — 2 ax”. 43. (1+x+xm) (3 − 2 x2 + xm) (5 xn−1 – 3 xm-1). 14. The converse of the Distributive Law evidently holds; that is, ab + ac ad = a(b+c-d), etc. = E.g., ax + bx= (a + b) x, 2 ay - 3 by (2 a 3 b) y. 15. If the coefficients of the multiplicand and multiplier, arranged to a common letter of arrangement, be literal, it is frequently desirable to unite the terms of the product which are like in this letter of arrangement. Ex. Multiply x + a by x + b. We have a + a x+b x2 + ax bx+ab x2 + ax + bx+ ab = x2 + (a + b) x + ab, by Art. 14. EXERCISES XII. Arrange the values of the following products to descending powers of That is, a product is 0 if one of its factors be zero. 17. The words is not equal to, does not have the same value as, etc., are frequently denoted by the symbol E.g., 72, read seven is not equal to 2. 18. It follows, conversely, from Art. 16: If a product be 0, one or more of its factors is 0. . Q=0 and P0; or P =0 and Q=0. EXERCISES XIII. 1. What is the value of 2 (a - b), when ba? 2. What is the value of (a + b)(c — d), when c = d? 3. What is the value of (b + c) (a + b − c), when c = a + b? - 4. What is the value of (x2 - 9) (x1 − 7 x3 + 2x-9), when x = 3? When x=-3? If P x Q x R = 0, what can we infer, 5. When P÷0? 6. When Q0? 7. When P÷0 and R ÷0? For what values of x does each of the following expressions reduce to 0: 8. x(x-2)? 9. (x −4)(x+7)? 10. (x-1)(x — a`? x(xa)(x − b)(x − e)? 1. One power is said to be higher or lower than another according as its exponent is greater or less than the exponent of the other. E.g., aa is a higher power than a3 or b2, but is a lower power than a or b. 2. Quotient of Powers of One and the Same Base. Ex. aa3 (aaaaaaa) ÷ (aaa) = = (aaaa) x (aaa) ÷ (aaa), by Assoc. Law, = aaaa = a* = a3¬3. This example illustrates the following principle: (i.) The quotient of a higher power of a given base by a lower power of the same base, is equal to a power of that base whose exponent is the exponent of the dividend minus the exponent of the divisor; or, stated symbolically, Express each of the following quotients as a single power: 3. Ex. 1. Division of Monomials by Monomials. 12 a ÷ 4 = 12÷4 xa, = 3 a. Ex. 2. - 27 x7 ÷ 3x2 = (−27 ÷ 3) × (27 ÷ x2), Ex. 3. 15 a3b2 ÷ (− 5 ab2) = [15 ÷ (− 5)] × (a3 ÷ =- - 3 a2. These examples illustrate the following method: The quotient of one monomial divided by another is the quotient of their numerical coefficients multiplied by the quotient of their literal factors. 18. x2-1y3m+2 by xn+1y2m-3 Simplify 20. a3x3÷(— ax3) × 2 axy. 11. 13. 3 ac. 11 a by- 5 a2. ab by 3 a2b. - 15 a5b7 by - 3 ab5. mon7p8 by — } m2n1po. 15. 25x2(x+1)3 by - 5x(x + 1)2. 17. - 27 xn+1y3m by - 9 xy2m. 19. an-1bn-2 by ɑn-3bn-4. 21. 35 x2y3z × 2 xz3 ÷ (7 x2y2z2). 22. a2n-16m+1cm+n ÷ anbmcm ÷ an−2bcn−3 ̧ 23. 6 xm+lyn-1÷(− xm-lym-n) × (3 x2y2z2). The Distributive Law for Division. 4. If the indicated operation within the parentheses in the quotient (86)÷2 be first performed, we have (8+6)÷214 ÷ 2 = 7. But if each term within the parentheses be first divided by 2 and the resulting quotients be then added, we have Therefore 8÷2+6÷ 2 = 4 + 3 = 7, as above. (8+6) ÷ 2 = 8 ÷ 2 + 6 ÷ 2. This example illustrates the following principle: Distributive Law. The quotient of a multinomial by a monomial is obtained by dividing each term of the multinomial by the monomial and adding algebraically the resulting quotients; that is, (a + b−c)÷d=a÷d+b÷d-c÷d. For, since dxd= =÷1, we have (a + b −c) ÷ d = (a ÷ d x d + b ÷ d × d - c÷d xd) ÷ d = (a ÷ d + b ÷ d - c÷d) × d÷d, by § 3, Art. 14, = a÷d + b÷d c÷d, since x d÷d = x 1. 5. It follows, conversely, from the Distributive Law that a÷d+b÷d-c÷d = (a + b−c)÷d. Zero in Division. 6. Since 0÷N=(aa)÷N, by definition of 0, = a÷N-a ÷ N= 0. We have 0 ÷ N = 0, when N 0. Observe that this relation is proved only when N 0. 7. It follows, conversely, from Art. 6: If a quotient be 0, the dividend is 0. That is, if M÷N= 0, then M = 0. Division of a Multinomial by a Monomial. 8. The division of a multinomial by a monomial is a direct application of the Distributive Law. Ex. 1. Divide 6x3- 12 x by 3x. We have (6x-12x)÷3x=6x+3x-12x + 3x =2x-4. Ex. 2. Divide - 105 a3b2-75 ab3 +27 ab by 15 ab. = − (− 105 a3b2) ÷ (− 15 a2b) — 75 a2b3 ÷ (− 15 a2b) +27 a2b1 ÷ (− 15 ab) =7ab5b2fb3. |