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1. Add 3a+b, 7a-5b, 7b-Sa, 4a-2b, and 8a-5b together. Result, 19a-4b.

2. Add 5ax+7bc-3a2, 8bc+4a2-2ax, 11a-4bc6ax and 2ax-3bc+6a2 into one sum.

Result, 8bc+18a-ax.

3. Add together the following quantities, 7abc+5a3d -3xy3, 2a3d+5xy-Sabc, 9bc3+8xy2+3aod, 4bc3abc -a'd, and a'd-10xy-abc--7bc3.

Result, 10a d+6bc3.

SUBTRACTION.

27. Change the signs of the quantities which are to be subtracted, or conceive them to be changed, then collect the terms as in addition.

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From 17ab-3xy+5b3

8ax-5bc 9aey-4bc

Take 11ab+4xy+7b2 7ax-7bc 2aey+7bc

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3. From 12a+10b+13ax-3ab, take 7a-5b+3ax.

Result 5a+15b+10ax-3ab.

4. From the sum of 3ab-7ax and 7ab+3ax, take 4ab-3ax-4xy. Result 6ab-ax+4xy.

5. From 5x-4xy+5, subtract 4x2-4xy+9.

Result x2-4.

6. From ax3-bx+x subtract px3-cx2+ex. Result, (a-p)3 +(c—b)x2+(1-e)x. 7. From bx + cx2-dx+e take px3-qx2+rx—s. Result, (b-p)3+(c+q)x3--(d+r)x+e+s.

MULTIPLICATION.

CASE 1.

28. When the factors are both simple quantities. Multiply the co-efficients together, and annex all the letters to the product.*

Note. When the signs of the factors are like, the pro duct must be made positive; when unlike, negative.

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* From. Def. 14, it is obvious, that the product of two or more powers of any letter, may be expressed by that letter with an index equal to the indices of the factors; thus, a3a3—aaa.aa=a5.

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29. When one of the factors is a compound quantity. Multiply the simple factor into each term of the compound one, and connect the products by their proper signs.

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30. When both factors are compound quantities. Multiply the whole multiplicand, by every term of the multiplier, and collect the several products as in addi

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a+b x2+xy+y3 x2+x3y+x2y2+xy3 +y*

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2. Multiply 3+3x2y+3xy2+y3 by x2+2xy+y3. Result, x+5x4y+10x3y2+10x2y3+5xy++y3. 3. Multiply 3x2+2xy+5y3 by x--xy+y2.

Result, 3x--x3y+6x2y3——3xy3+5y1.

4. Multiply 52+10xy+5y by 5x2--10xy+5y*.

Result, 25x-50x2y2+25y1.

5. Multiply a3+3a2b+3ab2+b3 by a3--3a3b+3ab3--b3. Result, a--3a+b2+3a2b±--b6.

6. Multiply a2--ab+b2-bc by a2+ab—b2+bc. Result, a-a2b2+2ab3+2b3c-b2c2—b4.

7. Multiply 4x3y+x2y2——xy3+y* by x2+xy+y2. Result, x+x+y3—x3y3+x2y1+y®.

8. Multiply x+y by x--y, and the product by x2+y2. Result, x*--y*.

9. Find the continued product of a2--2ab+b3, a2+ 2ab+b2, a3+ab+ab+b3 and a-b.

Result, a3--2aob2+2a2b¤——b3.

10. Find the continued product of 3x+6, 3x+2, Result, 81x-360x2+144.

3x-2 and 3x--6.

DIVISION.

CASE 1.*

31. When the divisor and dividend are both simple quantities.

Divide the co-efficient of the dividend by that of the divisor; expunge from the dividend such letters as are common to it and the divisor when they have the same exponents; when the exponents are not the same, subtract the exponent of the divisor from that of the dividend, and use the remainder as the index of the letter in the quotient; write in the quotient the letters which have not been expunged, with the co-efficient above determined.

Note. In division, as in multiplication, like signs in the divisor and dividend require the positive sign in the quotient; unlike signs, the negative.

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* If the student, who is just entering into this science, should express the powers by repetitions of the letters, the process will be more simple, and the grounds of the method indicated by the rule, be rendered obvious.

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