NOTE. The beginner must be careful to observe that in this process of multiplication the indices of one letter cannot combine in any way with those of another. Thus the expression 40a%b3c3 admits of no further simplification. 32. The general rule may now be given : RULE. To multiply two simple expressions together, multiply their coefficients, and add the indices of the like letters. The rule may be extended to cases where more than two expressions are to be multiplied together. Example 1. x3 x 23 x x8= x2+3 x 208 = x2+3+8 =x13. Example 2. 5x ́y3 x 8y2z5 x 3x24=120x%y%29. Note. The product of three or more expressions is called the continued product. MULTIPLICATION OF A COMPOUND EXPRESSION BY A SIMPLE EXPRESSION. 33. The symbol 4m means four times m, that is, m+m+m+m. So .: 10m=m+m+m+...... taking ten terms. Similarly am means that m is to be taken a times; that is am=ni+m+mt...... the number of terms being a. Again, (a+b) m=m+m+m+ taken a+b times = (m+m+m+. taken a tinies), together with (m+m+m+. taken b times) =am + bm .(1). Also (a-6) m=m+m+m+ taken a - b times =(m+m+m+...... taken a times), diminished by (m+m+m+. taken b times) (2). Similarly (u – b+c)m=am – bm +cm. 34. RULE. To multiply a compound expression by a single factor, multiply each term of the expression separately by that factor. Examples. (1) 3 (2a+36 – 4c)=6a+ 96 – 12c. =am- bm EXAMPLES V. a. Multiply together 1. 5.x2 and 7.25. 2. 4a3 and 5as. 3. Tab and 8a3b2. 4. 6xy2 and 5.33. 5. 8a3b and 65. 6. 2abc and 3ac3. 7. 2a383 and 24363. 8. 5aab and 2a. 9. 4a263 and 7a5. 10:5a4b3 and x-y. 11. 2°73 and 6a_24. 12. abc and XYZ. 13. 3a487,2,3 and 5a3ba.. 14. 4a’bx and 762x1. 15. 5a2x and 8cx. 16.5.203y3 and 6a’x3. 17. 2x2y and xby?. 18. 3a3x4yand a2yo. 19. ab + bc and a3b. 20. 5ab – 76x and 4a2b.2.?. 21. 5x+3y and 2x2. 22. a? +62 – C2 and a3b. 23. bc+ca - ab and abc. 24. 5a2+362 – 2c2 and 4a2bc3. 25. 5x2y + xy2 – 7x2y2 and 3.x3. 26.6.23 - 5.xoy+7.xya and 8.woy. 27. 6abc-7ab?cand a262, 28. 8a2b3 – şb2c3 and {ab. MULTIPLICATION OF COMPOUND EXPRESSIONS. 35. If in Art. 33 we write c+d for m in (1), we have (a+b) (c+d)=a (c+d)+b(c+d) =(c+d)a+(c+d) 6 [Art. 13.], =ac+ad+bc+bd ..............(3). Again, from (2) (a-6) (c+d)=a(c+d) - 6 (c+d) =(c+d) a-(c+d) . Similarly, by writing c-d for m in (1), (a+b)(c-d)=a{c-d)+b(c-d) =(c-d) a+c-db .(5). ....(4) =ac Also, from (2) =(c-da-(c-db (6). These results are very important. If we consider each term on the right-hand side of (6), and the way in which it arises, we find that tax+c= + ac. - bc. tax-d=-ad. The results enable us to state what is known as the Rule of Signs in multiplication. RULE OF SIGNS. The product of two terms with like signs is positive; the product of two terms with unlike signs is negative. 36. The rule of signs, and especially the use of the negative multiplier, will probably present some difficulty to the beginner. Perhaps the following numerical instances may be useful in illustrating the interpretation that may be given to multiplication by a negative quantity, - 3x +4=-3x4 = -3 taken four times 12. +3X -4 would thus seem to mean that +3 is to be taken -4 times ; this is at present without meaning, but we can easily give to it an interpretation which will be intelligible, and in harmony with what precedes. We have illustrated the difference between +4 and – 4, by supposing that +4 represents a line of 4 units measured in one direction, and - 4 a line of length 4 units measured in the opposite direction; so that, with reference to each other, each of the signs + and may be supposed to indicate a reversal of direction. Hence 3x –4 may be taken as indicating that 3 is to be taken 4 times, and further that the direction of the line which represents the product is to be reversed. Now 3 taken 4 times gives +12; and from this by changing the sign, which corresponds to a reversal of direction, we get – 12, Thus 3x --4= -12. Similarly - 3x -4 indicates that -3 is to be taken 4 times, and the sign changed; the first operation gives – 12, and the second + 12. Thus - 3X -4=+12. 37. We may here remark that we can give what meaning we like to any new symbol or operation, provided that we always employ such meaning, and that the meaning is not inconsistent with the fundamental principles of our subject. Multiplication by a negative quantity is a case in point; and the meaning we have attached to the operation is to proceed just as if the multiplier were positive, and then change the sign of the product. 38. To familiarize the beginner with the principles we have just explained we add a few examples in substitutions where some of the symbols denote negative quantities. Example 1. If a= -4, a=(-4)3 = - 4x - 4x - 4= - 64. Example 2. If a=-1, b=3, c= -2, find the value of – 3a4bc3. - 3abc3 = -3x (-1)* x 3 x (-2) = -3x1 x3 x(-8) If a= 15. 17. EXAMPLES V. b. -2, b=3, c=-1, x=-5, y=4, find the value of 1. 3a2b. 2. 8abc2. 3. – 563. 4. 6a2c2. 5. 4cg. 6. 3a2c. 7. - 62c2 8. 3a3c2. 9. -723bc. 10. – 2a4bx. 11. – 4a2c4 12. 303.23. 13. 5a2x2. 14. -74.cy. - Bax). 16. 405.03. - 5a2b2c2. 18. -7a3c3. 19. 84.23. 20. 7a5c4. If a=-4, b= -3,c=-1, f=0, x=4, y=1, find the value of 21. 3aR + bx – 4cy. 22. 2ab2 – 3bc2+2fx. 23. faz – 263 – ca. 24. 3a ́y3 – 5b2x – 2c3. 25. 203 - 363+7cy'. 26. 36%y* – 4b2f – 6cx. 27. 2.J(ac) – 31/(xy)+1(12c4). 28. 3/(acx) – 2.J(by) – 6/(c2y). 29. 7/(a.x) - 3./(6462)+5./fax). 30. 36J/(3bc) - 5/(4c4y3) - 2cy (360"). 39. The following examples further illustrate the rule of signs and the law of indices. Example 1. 4a2b x 3a462=-12a6b3. Example 3. Find the continued product of 3a2b, - 2a36", - ab“. 3a26 x – 2a3b2=-6a5b3, and - 6a573 x - ab4=6a6b7. Example 4. (6a3 – 5aab – 4ab2)x - 3ab3=-18a483 +15a3/4 +12a205. EXAMPLES V. c. Multiply together 1. ax and - 3ax. 2. - 2abx and – 7ab.x. 3. aềb and - ab2. 4. 6.c4y and -10xy. 5. - abcd and - 3a2b3cd5. 6. xyz and – 5.x2y3z. 7. 3xy +4yz and – 12xyz. 8. ab – bc and a2bc3. 9. - X - Y - z and - 3x. 10. 42 – 62 + c2 and abc. 11. - ab + bc- ca and - abc. 12. – 2a2b-4ab2 and - 7a26%. 13. 5x2y – Oxy2 + 8x-y2 and 3xy. 14. – 733y – 5xy3 and -- 8x373. 15. – 5xy-z + 3xyz2 – 8x2yz and xyz. 16. , 4.x2y2z2 – 8.xyz and – 12.xoyz3. 17. - 13.cy" - 15.cy and -7x*y. 18. Bxyz – 10x3y23 and – xyz. 19. abc - a2bc - ab?c and - abc. 20. - a2bc+baca – cab and - ab. Find the product of 21. 2a – 36+4c and - 3a. 22. 3x – 2y – 4 and – 5x. 23. şa- }h 24. 9a x2 - 2ax and - a x. 25. - 5ax2 and - 3a2 + ax – 222. 26. - xy and – 3.x2 +*xy. 27. - 8.0ya and - 4x2+2ya. 28. - 4x5y3 and 1.33 – 443. and fax. . 40. The complete rule for multiplying together two compound expressions may now be given. |