Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

If we examine the multiplicand and multiplier, with reference to a, we see that the product of 5a4b2 by a2b, must be irre ducible; also, the product of2ab3 by ab2. If we consider the letter b, we see that the product of ab1 by — ab2, must be irreducible, also that of 3a2b by a2b.

[ocr errors]

47. The following formulas depending upon the rule for multiplication, will be found useful in the practical operations of algebra.

Let a and b represent any two quantities; then a+b will represent their sum, and a b their difference.

I. We have

(a + b)2 = (a + b) × (a + b), or performing the multiplication indicated,

(a + b)2 = a2 + 2ab+ b2; that is,

The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second.

To apply this formula to finding the square of the binomial 5a2 + 8a2b,

we have (5a2 + 8a2b)2

Also,

= 25a4 +80a4b64a4b2.

(6a4b9ab3)236a8b2+108a5b1 + 81a2b.

II. We have, (a — b)2 = (a — b) × (a — b), or performing the multiplication indicated,

(a - b)2 = a2 — 2ab+b2; that is,

The square of the difference between two quantities is equal to the square of the first, minus twice the product of the first by the second, plus the square of the second.

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

III. We have

(a + b) x (a - b) — a2 — b2.
× =

by performing the multiplication; that is,

The sum of two quantities multiplied by their difference is equal to the difference of their squares.

To apply this formula to an example, we have

(Sa3 +7ab2) × (8a3 — 7ab2) = 64a6 — 49a2b1.

48. By considering the last thrée results, it is perceived that their composition, or the manner in which they are formed from the multiplicand and multiplier, is entirely independent of any particular values that may be attributed to the letters a and b, which enter the two factors.

The manner in which an algebraic product is formed from its two factors, is called the law of the product; and this law remains always the same, whatever values may be attributed to the letters which enter into the two factors.

DIVISION.

49. DIVISION, in algebra, is the operation for finding from two given quantities, a third quantity, which multiplied by the second shall produce the first.

The first quantity is called the dividend, the second, the divisor, and the third, or the quantity sought, the quotient.

50. It was shown in multiplication that the product of two terms having the same sign, must have the sign+, and that the product of two terms having unlike signs must have the sign. Now, since the quotient must have such a sign that when multiplied by the divisor the product will have the sign of the dividend, we have the following rule for signs in division.

[blocks in formation]

That is: The quotient of terms having like signs is plus, ana the quotient of terms having unlike signs is minus.

51. Let us first consider the case in which both dividend and divisor are monomials. Take

35a5b2c2 to be divided by 7a2bc;

The operation may be indicated thus,

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

Now, since the quotient must be such a quantity as multiplied by the divisor will produce the dividend, the co-efficient of the quotient multiplied by 7 must give 35; hence, it is 5.

Again, the exponent of each letter in the quotient must be such that when added to the exponent of the same letter in the divisor, the sum will be the exponent of that letter in the dividend. Hence, the exponent of a in the quotient is 3, the exponent of bis 1, that of c is 1, and the required quotient is 5a3bc.

Since we may reason in a similar manner upon any two monomials, we have for the division of monomials the following

RULE.

1. Divide the co-efficient of the dividend by the co-efficient of the divisor, for a new co-efficient.

II. Write after this co-efficient, all the letters of the dividend and give to each an exponent equal to the excess of its expo nent in the dividend over that in the divisor.

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

52. It follows from the preceding rule that the exact division of monomials will be impossible :

1st. When the co-efficient of the dividend is not divisible by that of the divisor.

2d. When the exponent of the same letter is greater in the divisor than in the dividend.

This last exception includes, as we shall presently see, the case in which the divisor has a letter which is not contained in the dividend.

When either of these cases occurs, the quotient remains under the form of a monomial fraction; that is, a monomial expression, necessarily containing the algebraic sign of division. Such expressions may frequently be reduced.

[blocks in formation]

Here, an entire monomial cannot be obtained for a quotient; for, 12 is not divisible by 8, and moreover, the exponent of c is less in the dividend than in the divisor. But the expression can be reduced, by dividing the numerator and denominator by the factors 4, a2, b, and c, which are common to both terms of the fraction.

In general, to reduce a monomial fraction to its lowest terms: Suppress all the factors common to both numerator and denomi

[blocks in formation]

12a8b6c7 3ab
402 3

6a3bc4d2 6a2d

7a2b

1

=

and.

=

14a3b2 2ab

In the last example, as all the factors of the dividend aro found in the divisor, the numerator is reduced to 1; for, in fact, both terms of the fraction are divisible by the numerator.

53. It often happens, that the exponents of certain letters, Are the same in the dividend and divisor.

[blocks in formation]

is a case in which the letter b is affected with the same exponent in the dividend and divisor: hence, it will divide out, and will not appear in the quotient.

But if it is desirable to preserve the trace of this letter in the quotient, we may apply to it the rule for exponents (Art. 51), which gives

[merged small][ocr errors]

The symbol 6o, indicates that the letter b enters 0 times as factor in the quotient (Art. 16); or what is the same thing, that it does not enter it at all. Still, the notation shows that b was in the dividend and divisor with the same exponent, and has disappeared by division.

In like manner,

15a2b3c2
3a2bc2

=5ab2c0 = 562.

54. We will now show that the power of any quantity whose exponent is 0, is equal to 1. Let the quantity be represented by a, and let m denote any exponent whatever.

[blocks in formation]
[blocks in formation]

1, since the numerator and denominator are equal:

[blocks in formation]

We observe again, that the symbol a° is only employed conventionally, to preserve in the calculation the trace of a letter which entered in the enunciation of a question, but which may disappear by division.

55. In the second place, if the dividend is a polynomial and the divisor is a monomial, we divide each term of the dividend by the divisor, and connect the quotients by their respective signs.

EXAMPLES.

Divide 6a2x4y6 — 12a3x3y + 15a4x5y3 by 3a2x2y2.

Ans. 2x2y4 - 4axy1 + 5a2x3y.

« ΠροηγούμενηΣυνέχεια »