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When one of the factors is a Compound Quantity.
Multiply every term of the multiplicand, or compound quantity, separately, by the multiplier, as in the former case ; placing the products one after another, with the proper signs; and the result will be the whole product required.
2. When two quantities are to be multiplied together, the result will be exactly the same, in whatever order they are placed; for a times c is the same as e times a, and therefore, when – a is to be multiplied by + c, or + c by- a. this is the same thing as taking-a as many times as there are units in tc; and as the sum of any number of nega. tive terms is negative, it follows that x + c, or taxmake or produce - ac.
3. When-a is to be multiplied by - c: here -a is to be subtracted as often as there are units in c: but subtracting negatives is the same thing as adding affirmatives, by the demonstration of the rule for subtraction ; consequently the product is c times a, or + ar.
Otherwise. Since a-a=0, therefore (a - a) X-c is also = 0, because 0 multiplied by any quantity, is still but 0 ; and since the first term of the product, or a x- cis = ac, by the second case ; therefore the last term of the product, or max c, must be tac, to make the sum = 0, Opac + ac = 0; that is, -ax
le + 8xy
222 30 + 5
4x - 0 + 3ab 4a
CASE III. When boin the Factors are Compound Quantities ; MULTIPLY every term of the multiplier by every term of the multiplicand, separately, setting down the products one after or under another, with their proper signs; and add the several lines of products all together for the whole product required. a + 6 3x + 2y
2x3 + xy2ye a to 4.2 - 54
3x-3y 2% + ab
6.33+ 3.r@y-hy? + ab + 62 - 15x7 - 10y2 - 6x2y—3.ry2 +6y
3 + x 4xy
al + 2ab + 62
1232 - 7xy - 10ye 6x3_3.ry-9xy?+ 6ys
Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them, begin at the left-hand side, and multiply from the left hand towards the right, in the manner that we write, which is contrary to the way of multiplying nunbers. Bu in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lines above, when there are such like quantities ; which is the easiest way for adding them up together.
in many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum with a sign of multiplication between them. As(a + b) (a - b) * 3ab, or a +6 a-6. 3ab.
EXAMPLES FOR PRACTICE.
1. Multiply 10ac by 2a.
Ans. 2092C. 2, Multiply 3a2 26 by 36.
Ans. 9a2b-66%. 3. Multiply 3a + 26 by 3a
Ans. 9a? -46%. 4. Multiply x?- xy + yo by x + y. Ans x3 + ys. 5. Multiply a3 + a2b + ab? + b3 by a b. Ans. 24-64. 6. Multiply a2 + ab +62 by a? - ab + 6%. 7. Multiply 3x2 - 2xy + 5 by x2 + 2xy 6. 8. Multiply 3a? - 2ar + 5x 2 by 3a2-4ax-7x2. 9. Multiply 3.03 + 2x?y2 + 3y3 by 2x33xy? + 3y 3, 10. Multiply ad + ab + 62 by a 26.
Division in Algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left-hand side, and dividing all the parts of the dividend by the divisor, when they can be so divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases.
When the Divisor and Dividend are both Simple Quantities ;
Set the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. Then abbreviate these terms much as can be done, by cancelling or striking out all the
atters that are common to them both, and also dividing the one co efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common measure.
Note. Like signs in the two factors make + in the quotient; and unlike signs make-; the same as in multiplication*
1. To divide bab by 3a,
баб Here 6ab + 3a or 3a) 6ab ( or
_= 26. 3a
abx 2. Also cC =1; and abxbxy :
bxy y 3. Divide 1602 by 8.0
Ans. 2x. 4. Divide 12u« c2.by — 3a2x.
Ans. 41. 5. Divide 15ay? by 3ay.
6. Divide - 180x2y by- Saxz.
* Because the divisor multiplied by the quotient, must produce the dividend. Therefore,
1. When both the terms are t, the quotient must be t; because + in the divisor x t in the quotient, produces t in the dividend.
2. When the terms are both, the quotient is also t; because -in the divisor x + in the quotient; produces - in the dividend.
3. When one term is + and the other —, the quotient must be ; because + in the divisor x in the quotient produces in the divia dend, or in the divisor x.t in the quotient gives in the dividend.
So that the rule is general; viz, that like signs give to and unlike signs give, in the quotient.
When the Divdiend is a Compound Quantity, and the Divisor
a Simple one :
* Divide every term of the dividend by the divisor, as in the former case.
atb 1. (ab +62) 2b, or
= a + b. 26
10ab + 150x 2. (10ab + 15ax) 5a, or
2b + 33 5a
30az 48z 3. (30az - 48z) z, or
When the Divisor and Dividend are both Compound Quantities:
1. Set them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower.
2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the quotient.
3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend.
4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before ; and so on to the end, as in common arithmetic.