17.8 x 7k; 56 k÷7 k; 56k ÷ 8. 8a 18. 4 x (-2a); −8 a÷(−2 a); -8a÷4. 19. 7 x(-3a); - 21 a÷(-3 a); — 21 a ÷ 7. 20. − 8x7t; – 56t =(−8); — 56 t + 7 t. - 21. -2x(-5r); 10r÷(− 2); 10r÷(− 5 r). 22. 3 ax(-2 a); — 6a2 ÷ 3a; -6a2÷(-2 a). 23. 21k÷(-3k); -8 a2÷2a; - 18 x ÷ (− 6x). 145. Integral Algebraic Expression. An algebraic expression is an integral algebraic expression if there are no literal numbers in a denominator. 146. The Law of Signs in Division. The student should remember that in dividing one number by another: 1. The quotient of two numbers having like signs is positive. 147. The Law of Exponents in Division. Since a3 · a2 = a5, therefore a3 ÷ a2 = = a3 or a5-2, and a5a3: = a2 or a5-3 Similarly. a8 · a3 = all, ... all ÷ a8 = In general, a3 or all-8. and all a3 = a8 or all-3. am÷an= am-n. The equation am ÷ an = am-n, gives in algebraic symbols, the law of exponents in division. In words, this law may be stated: The exponent of any base in the quotient is equal to its exponent in the dividend minus its exponent in the divisor. = 5. a3÷aka2k. (Why?) = - k2 = k1. (Why?) 6. ar+2 ÷ a′ = ar+2¬r = a2. 149. State the definition of division. Define dividend, divisor, and quotient. (§§ 51, 142.) 8 a2b3c÷(−2 a2b) = — 4 b2c is an immediate consequence of the definition of division since (-2 a2b) x (-4b2c)=8 a2b3c. ORAL EXERCISE 150. Using only the definition of division give answers to the following and explain: 151. When the examples are simple, the definition of division along with our previous practice in multiplication will enable us to find the quotients. A rule can be stated, however, that will help us to perform divisions. To divide a monomial by a monomial : 1. Divide the numerical coefficient of the dividend by that of the divisor, keeping in mind the law of signs. 2. Subtract the exponent of any letter in the divisor from the exponent of that letter in the dividend to find its exponent in the quotient. 3. Omit from the quotient any letter whose exponent in the dividend is the same as its exponent in the divisor and write unchanged in the quotient any letter that occurs only in the dividend. Why is the sign of the quotient negative? How is the literal part of the answer obtained? 2. 15 c3d3f+(-5cd3f)=3c. Explain. 3. 2n+42n−2 =2n+4−(n−2) = 26 = 64. 4. — 3x5 (α — b)1 ÷ x2 (a — b) — — 3 x3 ( a − b)3. 152. Find the quotients: EXERCISE 9. 7a2b2c÷(- 8 abc). 10. 5040 аbo ÷ 720 ab2. 33x3y 31. 15 m2 (x2-1)4 ÷ 3 m (x2 – 1)3. 32. 15 cd45 d3 + 22 c5d8 x cd ÷ 11c4d8. 33. 10 a2b × ab2 ÷ 5 ab – 5 aab3 ÷ ( − 5 a2b). 34. aman. 35. 2am a”. 36. am+n÷a”. 37. a2na". 38. 33 as+2b3 ÷ 3 a3b. 39. 28 a +263 +(-4a2b). DIVISION OF A POLYNOMIAL BY A MONOMIAL 153. 1. Since 2 x (a + b)=2a+2b, therefore (2a+2b)÷ 2 = a + b, by the definition of division. 2. Since a (x + y) = ax + ay, therefore (ax +ay) ÷ a = x + y. 3. (2x+4y)+2=x+2y. CHECK. 2(x+2y) = 2x+4y. 4. (4 a2 + 10 a) ÷ a = 4a+10. 5. (5a10b+ 15 c)+(-5)=a+2b-3c. ORAL EXERCISE 154. Find the quotients: 11. (ax + bx + cx)÷x. 20. (a+1-a")÷a". 155. From these examples we derive the following rule: To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and unite the results with their respective signs. The simplest verification of such exercises is by using the relation, divisor × quotient = dividend. (dx q=D.) |