96 -3x3m+-12cm3 x-y 15b+6x+3y 11. What is the sum of 3ag+6am-9xy+3ab−xy+4ag +10am-7xy-6ab+5xy+4ag-13am? Ans. 11ag+3am-12xy-Sab. 12. What is the sum of 4a3x-5a3y+7am-3a3x -10a3y-4am +9a3y-7a2x-13am + 6a3y-11a2x+am -a3y+a3y-6a2x? Ans. -23a2x-9am. 13. What is the sum of -3xy+5n+3ax-10am-6xy+7n -4ax+8am-2xy+10n+6am-4ax? Ans. 11xy+22n-5ax+4am. 14. What is the sum of 4a2+5a2b2c2-9a2+6a2b2c2 +10a3x+7a3x+8a2-13a2b2c2+5a2-3a3x+3a2b2 c2? Ans. 8a2+a2b2c2+14a3x. 15. What is the sum of √a-3xy—3⁄4 m−n+6xy+5 √ȧ +7n-7xy+9√a-73m+16n-5√a? Ans. 10a-4xy-8m+22n. 16. What is the sum of 6a3√ b+5x2y*—7x+5a3b3+6x —3a3b*—4x2+vy+2a3 √b—10x+8a*b*? Ans. 18a3b+x2y +x2y-11x. 17. What is the sum of 3 √a+b−5 √x+5x2y3+7√a+b +3x2y3 −7√x+√a+b−8x2y3+2√x+10√a+b? Ans. 21 va+b-10 √x. 18. What is the sum of 5a3b3c-4ab2c3+2a2b2 c2 —7ax +6a3b2c-5ab2c3-13ax-7a2b2c2+3ab2c3-8a3b2c? Ans. 3ab2c-6ab2c3-5a2b2c2-20ax. SUBTRACTION. (26.) SUBTRACTION, in Algebra, is the finding the simplest expression for the difference of two algebraic expressions. (27.) If we subtract the positive quantity b from a, we obviously obtain a-b, which is the same as the addition of a and -b. Again, if we wish to subtract b-c from a, we obtain by subtracting b from a, a-b, but we have subtracted too much by the quantity c, therefore adding c, we get a-b+c, which is the same as the addition of a and b+c. From this, we see that subtracting a quantity is the same as adding it after the signs are changed. Hence, for the subtraction of algebraic quantities we have this RULE. I. Write the terms to be subtracted under the similar terms, if there are any, of those from which they are to be subtracted. II. Conceive the signs of the terms, of the polynomial to be subtracted, to be changed, and then proceed as in addition. 1 5. From -4my. 3a2bc-7axy+3my+a take a2bc+8axy+6a Ans. 2a2bc-15axy+7my-5a. 6. From 8ab√c-12a3b+6cx-7xy take 9ab√c-13a3b +8xy-an+3cx. Ans. —ab√c+a3b+3cx−15xy+an. 7. From 15a3x-14a2y+3ab5+6amn take 6mg+3a -5a3x-7a2y+3ab5—4amn-4. 8. From Ans. 20a3x-7a2y+10amn-6mg-3a+4. 13a1x3y+3ax-7ab+6mg-x2y2 take 5xy +4a1x3y-6ax+9ab+2mg+5x2y2. Ans. 9a1x3y+9ax-16ab+4mg-6x3y2—5xy. 9. From 7a2b+3a2-4b1x—3axy+4x2—3xy3 take 3ab Ans. 12a2b-a2+3b1x-3axy+4x2-6xy3 —3ab+17. 10. From 4a3bx-7axy+3p2q+17-x take 4p2q-13 +7a3bx+8axy-7x+3f—mg+n. Ans. -3a3bx-15axy-p2q+30+6x-3f+mg―n. 11. From 6am+x take 3am+y. Ans. 3am+x-y. 12. From 3a2m-6x2y3+2xy take 4a2m+6x2y3+5xy. Ans. -a2m-12x2y3 —3xy. 13. From 3amx−43+x−y+27d take 15n+6g-3+4y -8d+7amx-x+pq¬rs+g. Ans. —4amx-40+2x-5y+35d-15n-7g-pq+rs. 14. From a+b take a-b. Ans. 2b. (28.) We can express the subtraction of one polynomial from another, by writing the polynomial which is to be subtracted, after enclosing it within a parenthesis, immediately after the other polynomial from which it is to be subtracted, observing to place the negative sign before the parenthesis. ab-6xy+3am―(4ab+3xy+am) Thus, denotes, that the polynomial enclosed within the parenthesis is to be subtracted from the one which precedes it; and since, by (Art. 27), to perform subtraction, we must change all the signs of the terms to be subtracted, we may remove the parenthesis provided we change the signs of the terms which it encloses and conversely, we may enclose any number of terms within the parenthesis, with a negative sign before it, if we observe to change the signs of the terms thus enclosed. In this way we can transform the expression |