CHAPTER IV. SUBTRACTION. 26. THE simplest cases of Subtraction have already come under the head of addition of like terms, of which some are negative. [Art. 19.] and Also, by the rule for removing brackets [Art. 22.], 3a-(-8a)=3a +8a = 11a, -3a-(-8a)=-3a + 8a =5a. SUBTRACTION OF UNLIKE TERMS. 27. The method is shown in the following example. =4a-3b+5c-3a+2b+c -4a-3a-3b+2b+5c+c =a-b+6c. It is, however, more convenient to arrange the work as follows, the signs of all the terms in the lower line being changed. RULE. Change the sign of every term in the expression to be subtracted, and add to the other expression. NOTE. It is not necessary that in the expression to be subtracted the signs should be actually changed; the operation of changing signs ought to be performed mentally. Example 1. From 5x2+xy - 3y2 take 2x2+8xy — 7y2. 5x2+ xy-3y2 3x2 - 7xy+4y2. Example 2. Subtract 2x4 -3x2+7x-8 from x1−2x3 -- 9x+4. 9. ab + cd-ac-bd from ab + cd + ac+bd. 10. - abcd-ac+bd from ab-cd+ac-bd. 11. From 3ab5cd-4ac-6bd take 3ab+6cd - 3ac-5bd. 12. yz-zx+xy take - xy+yz — ZxX. 13. - 2x3- x2 - 3x+2 take x3 − x + 1. 14. - 8x2y+15xy2+10xyz take 4x2y - 6xy2 – 5xyz. 15. a-b+c take ja+3b-c. 16.xyz take 3-by-. From EXAMPLES IV. b. 1. 3xy-5yz+8zx take - 4xy+2yz - 10zx. 2. 3. 4. 5. -8x2y2+15a3y+13xy3 take 4x2y2+7x3y-Sxy3. a2bc+b2ca + c2ab take 3a2bc-5b2ca-4c2ab. -7a2b+8ab2+ cd take 5a2b - 7ab2+6cd. 6. 8x2y+5xy2 - x2y2 take 8x3y — 5xy2+x2y2. 7. 10a2b2+15ab2+8a2b take - 10a2b2+15ab2 – 8a2b. 8. 4x2-3x+2 take - 5x2+6x−7. 9. x3+11x2+4 take 8.x2 – 5x – 3. 10. 11. −8α2x2+5x2+15 take 9a2x2 - 8x2 - 5. Subtract x3-x2+x+1 from x3+x2-x+1. 12. 3xy2 -3x2y+x3 −y3 from x3+3x2y+3xy2+y3. 13. b3+c3-2abc from a3+b3 - 3abc. 14. 7xy2 — y3 – 3x2ÿ+5x3 from 8x3+7x2y — 3xy2—y3. 15. x2+5+x-3x3 from 5xa – 8x3 – 2x2+7. 16. 17. 18. a3+b3c3-3abc from 7abc-3a3+5b3 — c3. 1−x+x5 - x - x3 from x2-1+x-x2. 7a1-8a2+3a5+ a from a2-5a3-7+7a5. 19. 10a2b+8ab2 – 8a3b3 — b4 from 5a2b - 6ab2 - 7a3b3. 20. a3-b3+8ab2-7a2b from -8ab2+15a2b+b3. 28. We shall close this chapter with an exercise containing miscellaneous examples of Addition and Subtraction. EXAMPLES IV. c. 1. To the sum of 2a-36-2c and 2b-a+7c add the sum of a-4c+7b and c-6b. 2. From 5x + 3x - 1 take the sum of 2x-5+7x2 and 3x2+4-2x2+x. 3. Subtract 3a-7a3+5a2 from the sum of 2+8a2-a3 and 2a3-3a2+a- 2. 4. Subtract 5x2+3x-1 from 23, and add the result to 3x2+3x-1. 5. Add the sum of 2y-3y2 and 1-5y3 to the remainder left when 1-2y2+y is subtracted from 5y3. 6. Take x2-y2 from 3xy-4y2, and add the remainder to the sum of 4xy - x2 - 3y2 and 2x2+6y2. 7. Find the sum of 5a −7b+c and 36 – 9a, and subtract the result from c-4b. 8. Add together 3x2-7x+5 and 2x2+5x-3, and diminish the result by 3x2 + 2. 9. What expression must be added to 5x2 - 7x+2 to produce 7x2-1? 10. What expression must be added to 4x3-3x2+2 to produce 4x+7x-6? 11. What expression must be subtracted from 3a − 5b+c so as to leave 2a - 4b+c? 12. What expression must be subtracted from 9x2+11x-5 so as to leave 6x2-17x+3? 13. From what expression must 11a2-5ab–7bc be subtracted so as to give for remainder 5a2+7ab+7bc? 14. From what expression must 3ab+5bc - 6ca be subtracted so as to leave a remainder 6ca - 5bc? 15. To what expression must 7x3- 6x2 - 5x be added so as to make 93 - 6x - 7x2? 16. To what expression must 5ab - 11bc - 7ca be added so as to produce zero ? 17. If 3x2-7x+2 be subtracted from zero, what will be the result? 18. Subtract 3x3-7x+1 from 2x2 - 5x-3, then subtract the difference from zero, and add this last result to 2x2 - 2x3 – 4. 19. Subtract 3x2-5x+1 from unity, and add 5x2 - 6x to the result. CHAPTER V. MULTIPLICATION. MULTIPLICATION OF SIMPLE EXPRESSIONS. 29. WHEN there is no sign between symbols or expressions, it is understood that the symbols or expressions are to be multiplied together. Thus ab=axb. 3ab=3x axb. a(x-y)=ax(x-y). (x+y) (a+b) is the product of x+y and a+b. In Algebra, as in Arithmetic, the product is the same in whatever order the factors are written. [Art. 13.] 31. When the expressions to be multiplied together contain powers of different letters, a similar method is used. Example. 5a3b2 × 8a2bx3=5aaabb × 8aabxxx =40ab3ე3. |