Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

3d, If both terms be affected with the sign, the product must have the sign +·

Or in more general terms, If both terms have the same sign, whether + or, the product must have the sign+, and if they have different signs, the product must have the sign.

27. Multiply 3a2b—2ac+5

by

Product

7ab-2ac-1.

21 a3 b2-14 a2bc +35 ab

6 a3 b c + 4a2 c2 — 10 a c -3ab2ac-5.

21 a3b-14 abc +35 ab-6 a3b c +4 ac2-8 ac-3ab-5.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

It is generally much easier to trace the effect produced by each of several quantities in forming the result, when the operations are performed upon letters, than when performed upon figures. The following are remarkable instances of this. They ought to be remembered by the learner, as frequent use is made of them in all analytical operations.

Let a and b represent any two numbers; a +b will be their sum and a b their difference.

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

That is, if the sum and the difference of two numbers be multiplied together, the product will be the difference of the second powers of these two numbers.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

That is, the product of the sum of two numbers, by itself, or the second power of the sum of two numbers, is equal to the sum of the second powers of the two numbers, added to twice the product of the two numbers.

[blocks in formation]

The answer is a2.

[ocr errors]

2ab+b, which is the same as the last, except the sign before 2 a b.

Multiply a +2 a b + b2 by a + b, that is, find the third

power of a+b.

Ans. a3 +3ab + 3 a b3 + b3

This is expressed in words thus: the third power of the first, plus three times the second power of the first into the second, plus three times the first into the second power of the second, plus the third power of the second.

Multiply a-2 ab-b

by a-b.

Ans. a3-3 ab + 3 a b2 —b3.

Which is the same as the last, except the signs before the second and last terms.

Instances of the use of the above formulas will frequently occur in this treatise.

Division of Algebraic Quantities.

XIV. The division of algebraic quantities will be easily per formed, if we bear in mind that it is the reverse of multiplication, and that the divisor and quotient multiplied together must reproduce the dividend.

The quotient of a b divided by a is b, for a and b multiplied together produce a b. So ab divided by b gives a for a quotient, for the same reason.

If 6 a b c be divided by 2 a, the quotient is 3 b c.

If

If

If

If

If.

by 2b, the quotient is 3 a c.

by 2 c, the quotient is 3 a b.

by 2b, the quotient is 2 a c.

by 3 ab, the quotient is 2 c.
by 6 a the quotient is bc.

For in all these instances the quotient multiplied by the diviscr, produces the dividend 6 abc.

Examples.

1. How many times is 2 a contained in 6 abc? Ans. 3 bc times, because 3 b c times 2 a is 6 abc.

2 lf ô a b c be divided into 2 a parts, what is one of the paris ?

Ans. 3bc; because 2 a times 3 b c is 6 a b c.

Hence we derive the following

RULE. Divide the coefficient of the dividend by the coefficient of the divisor, and strike out the letters of the divisor from the divi

dend.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Observe that 4 a is the same as 4 a a a and a2 is the same as a a; 4 a a a divided by a a gives 4 a for the quotient.

It was observed in multiplication, that when the same letter enters into both multiplier and multiplicand, the multiplication is performed by adding the exponents, thus a multiplied by a is a3+= a3. In similar cases, division is performed by subtracting the exponent of the divisor from that of the dividend. a' divided by a2 is a

==

10. Divide

2

= a3.

[blocks in formation]
[merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

The division of some compound quantities is as easy as that

of simple quantities.

If a+b+c be multiplied by d the product is

d(a+b+c) or ad+bd+cd.

Therefore if a d+bd+cd be divided by d, the quotient is a+b+c.

If ad+bd+c d be divided by a+b+c, the quotient is d. When a compound quantity is to be divided, let the quantity, if possible, be so arranged that the divisor may appear as one of the factors, and then that factor being struck out, the other factor will be the quotient.

19. Divide 12 a2b-9 ac by 3 a.

12 ab-9ac3a (4ab3c)

Ans. 4 ab-3°c.

Observe that a is a factor of both terms, and also 3. Hence the quantity 12 ab-9 a c, can be resolved into factors; thus 3 (4 ab-3 a c), or a (12 a b-9 c), or 3 a (4 ab-3c). In the last form the divisor 3 a appears as one factor, and the other factor 4 ab-3 c is the quotient.

Note. Any simple quantity, which is a factor of all the terms of any compound quantity, is a factor of the whole quantity; and that factor being taken out of all the terms, the terms as they then stand, taken together, will form the other factor.

20 Divide 8 a2b3 16 a3 bc by 2ab4 a2 c.

a3 =

8 a b3-16 a' b2 c 4 a b3 (2 ab- 4 a2 c.)

21. Divide

Ans. 4 ab

3 a b c — 15 a b3d+9 a3b d2 by 3a b.

22. Divide 15 a b c 30 ac2 + 25 a3 c d

[blocks in formation]

23. Divide 36 a12 ba c— 28 a11 ba c2 + 40 a® b® c3

by

α

9 ao 7 a' b2 c + 10 a2 b'c2.

24. Divide 42 a84 a1 bc

by 1-2 a'b'c...

Algebraic Fractions.

XV. When the dividend does not contain the same letters as the divisor, or but part of those of the divisor, the division cannot be performed in this way. It can then only be expressed. The usual way of expressing division, as has already been explained, is by writing the divisor under the dividend in the form of a fraction. Thus a divided by b is expressed

a

« ΠροηγούμενηΣυνέχεια »