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and unlike quantities are expressions which contain different letters or different powers of the same letters; thus 3a2c3 and 7a2c3 are like quantities, whilst 5bx2y and 962xy2 are unlike.

17. A simple quantity consists of one term, as 4cx; a compound quantity consists of two or more terms connected by the signs or; thus 16a2c+ab and 13c2x3-4cd are compound quantities.

18. A vinculum, bar —, or parenthesis (), is used to collect several quantities into one; thus a+xd or (a+x)d denotes that the sum of a and x is to be multiplied into d; also 4ac-b2 or (4ac-62) indicates the square root of the remainder left by subtracting the square of b from four times the product of a multiplied into c.

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19. The reciprocal of a quantity is the quotient arising from dividing unity by that quantity; thus is the recipro

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or", where n may represent any number either whole or fractional, and is used as a general symbol for any expo

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20. Find the numerical values of the following expressions, when a=8, b=4, c=3, d=2, e=15, ƒ=0.

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In the following equations, the first and second sides will always give the same numerical value, if the same value be given to the letters on each side: verify this.

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8. (a+x)(a-x)=a2x2.

9. (a+b—c) (a—b+c)—a2—b2—c2+2bc. 10. x-y=(x—y)(x3 + x2y+xy2+y3).

ADDITION

21. Is commonly divided into three cases:- -1st, When the quantities are like, and have like signs; 2d, When the quantities are like, but have unlike signs; 3d, When the quantities are not all like, and have unlike signs.

CASE I. RULE.-Add the coefficients together, and to their sum annex the literal part.

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Ex. 1. Find the sum of ax+ax2+3ax.

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Ans. 9ax.

2. Find the sum of 2amx3 +4amx3 +7amx3 +44аmx3.

Ans. 17 amx3.

3. Find the sum of 4√x2+y4+7√x2+y1+√x2+y1 +3√x2+y2+19√x2+ y2+10√x2+ya.

Ans. 44/x+y1.

4. Find the sum of 12(x1—2)3⁄4+(x1—2)3 + 7 (x1a—2)3 +4(xa—2)3+94(x+—2)3+81(x^—z)3.

Ans. 411⁄2(æ1—z)3.

22. CASE II. When the quantities are like, but have unlike signs.

RULE. Add the coefficients of the plus quantities into one sum, and those of the minus quantities into another, their difference is the coefficient of the sum, and is plus if the sum of the plus coefficients be the greater, and minus if it be the less.

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Ans. 4a/mx.

1. Find the sum of amx-4a/mx+7a√mx—3a/mx

+4a√mx——√mx.

2. Find the sum of 7mpx2y3-4mpx2y3-8mpx2y+ 4mpx2y5-6mpx2y3+2mpx2y3.

Ans. -5mpx2y3.

3. Find the sum of 3(x2—y?)*+7(x2—y?)*—4(x2_ y?)*_8(x2—y?)$+5(x2—y2)*_2(x2—y?)$.

Ans. (x2—y2)1 4. Find the sum of 5(b—c)x3_7(b—c)x3+4(b—c)x3 —4(b—c)x3 +7(b—c)x3—3(b—c)x3.

Ans. 2(b—c)x3·

23. CASE III. When both the signs and the quantities are unlike, or some like and others unlike.

RULE. Find the sum of each parcel of like quantities by the last rule, and write the several results after each other, with their proper signs.

NOTE. The above rules will be obvious from the following considerations:-The first rule is simply this, that any number of quantities, as 4, and 5, and 7, of the same kind, will make 4+5+7, or 16 quantities of the same kind. In the second, it must be remembered, that minus quantities are subtractive, while plus ones are additive, and that to add and then subtract the same quantities, is the same as to perform no operation at all; therefore to add a greater quantity, and then subtract a less, is the same as to add their difference; and to subtract a greater, and then add a less, is the same as to subtract their difference, which is the rule. Again, it is evident that the third rule just enables us to combine several accounts in the second into

one sum.

EXAMPLE.

3xy +4az-3cx -4ax2+4xy-3az 7cx-3αx2-2xy -5az +4cx —ɑx2 3xy +5cx +3αx2

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8xy-4az+13cx-5αx2

Here we begin with the first term, which contains xy. We therefore collect all the (xy)'s into one sum; then find the sum of the (az)'s, and so on till we have collected all the terms.

1. Find the sum of 4x2+8xy+4y2, 3x2+4xy + y2, 2x2-4xy+2y2, and x2-y2.

Ans. 10x2+8xy+6y2. 2. Find the sum of 3ab+4ac-cd, -6bc+8cd+4ab, +6bc-4ab+5cd, and 4bc+2ab.

Ans. 5ab4ac+12cd+4bc. 3. Find the sum of 4a2+10ab+762, 3a2+4ab+b2, 2a2+6ab+462, and a2-6ab-762. Ans. 10a2+14ab+562.

4. Find the sum of -7a14ab36, and a—4ab4b.

+2b, 3a-4ab2b,

Ans. 2a+3a+26.

SUBTRACTION.

24. RULE. Change the signs of the quantities to be subtracted, or suppose them changed, and then collect the quantities, as in addition.

NOTE. This rule will appear evident from the following considerations :-If we are required to subtract 3a-b from 5a, this means that we are to take away (3a—b) from 5a; if then we take away 3a, the remainder will be 5a-3a; but it is evident we have here taken away too much by b; we must then add b, and we will have for the true remainder 5a—3a+b=2a+b, where it is evident that the signs of the subtrahend have been changed, and then the quantities collected by addition. The same process of reasoning will apply to any

other case.

EXAMPLE.

From 7a+5ac2-3bc-4bc2,
Take 2a-3ac2+4bc-9bc2.

Rem. 5a+8ac2—7bc+5bc2.

1. From a2+2ab+b2, take a2-2ab+b2.

2. Fromx+3x2y+3xy2+y3, take x3-3x2y+3xу2—y3.

Ans. 4ab.

Ans. 6xy+2y3.

3. From 3-a3, take a2-2ax+x2.

Ans. x3-x2+2ax—a2—a3.

4. From 3bx2-4cx+5x3, take 5x3-4bx2+3cx.

Ans. 7bx2-7cx.

MULTIPLICATION.

25. In algebraic multiplication three things are to be attended to; first, the sign; second, the coefficient; and third, the literal part of the product.

RULE. When the signs of the factors are like, the sign of the product is plus, and when the signs are unlike, the sign of the product is minus. The product of the coefficients of the factors is the coefficient of the product. And the letters of both factors, written after each other as the letters of a word, form the literal part of the product, the letters being commonly arranged in the order of the alphabet.

NOTE 1. That like signs give plus, and unlike give minus, can be shown in the following manner :-When +b is to be multiplied into +a, the meaning is, that +b is to be added to itself as often as there are units in a, and therefore the product is +ab; if now (b-b), which is evidently =0, be multiplied by a, the product must be =0; but +bx+a has been shown to be +ab, and that the whole product may be =0, the other part must be ab; therefore -bx+a=-ab. Again, since the product of two factors is the same, whichever be considered as the multiplier, +ax--b——ab, and if (a—a), which equals 0, be multiplied into b, the product must be 0; but it has been shown, that +ax-b=-ab, therefore that the product may be 0, -ax-b must be equal to +ab. Hence like signs give plus, and unlike signs give minus.

NOTE 2. When the same letter appears in the multiplicand and multiplier, it will appear in the product with a power equal to the sum of its powers in each factor; for a3xa2 denotes (aaa)×(aa)= aaaaa=a3—a(3+2), that is, its power in the product is the sum of its powers in the multiplier and multiplicand.

Multiplication naturally divides itself into three cases. 26. CASE I. When the multiplicand and multiplier are both simple quantities.

RULE. Multiply the coefficients together for the coefficient of the product, and the letters together for the literal part, and prefix the proper sign. EXAMPLE. 5ac X3ab2-15a2b2c. 1. Multiply 3a2bc, by 7ab2c3. 2. Multiply-5acd, by 3bcd. 3. Multiply -4a2c3, by —7bcd. 4. Multiply 7acx, by -3acy.

Ans. 21a3b3c4.

Ans.

15abc2d2.

Ans.

28a2bc4d.

Ans. -21a2c2xy.

27. CASE II. When the multiplicand is a compound and the multiplier a simple quantity.

RULE. Multiply each term of the multiplicand by the multiplier, and write the several products after each other, with their proper signs.

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