In multiplying one polynomial by another, there are always two terms of the total product which are not produced by the reduction of similar terms in the partial products. These two terms are the term affected with the highest exponent of any letter, and the term affected with the lowest exponent. If the terms of the multiplicand, multiplier, and product be arranged in the order of the powers of some letter,* as is usual, and as may be seen in the above examples, then the two terms in question of the product will be the first and last, the one being produced by the multiplication of the first of the multiplicand by the first of the multiplier, and the other by the multiplication of the last of the multiplicand by the last of the multiplier. The first of the multiplicand by the second of the multiplier usually produces a term similar to that which is produced from the multiplication of the second of the multiplicand by the first of the multiplier. The same is the case with the first and third of each, the first and fourth, the second and fourth, the third and fourth, and so on. When a polynomial, arranged according to the powers of some letter, contains many terms in which this letter has the same exponent, these terms, after suppressing from them the letter of arrangement, may be placed in a parenthesis, or in a vertical column with a vinculum placed vertically on the right, and the letter of arrangement, with its proper exponent, following after. The polynomial in the parenthesis, or vertical column, is to be regarded as the coefficient of the power of the letter which follows, and is to be operated with exactly as we do with a numerical coefficient; i. e., multiply the coefficient of the letter of arrangement in the multiplicand by the coefficient of the same letter in the multiplier, and afterward add the exponents of this letter. * The letter chosen for this purpose is called the letter of arrangement. (16b4-8b3 (4b-1) a3 — (16b3-4b2b+2)a2+(32b1-8b3-4b+2b-1)a-(3265-16b-8b+46) +2b-1)a—(32b5—16b1—8b3+46o) MULTIPLICATION BY DETACHED COEFFICIENTS. 14. In many cases the powers of the quantity or quantities in the multiplication of polynomials may be omitted, and the operation performed by the coefficients alone; for the same powers occupy the same vertical columns, when the polynomials are arranged according to the successive powers of the letters; and these successive powers, generally increasing or decreasing by a common difference, are readily supplied in the final product. Since xxxx, the highest power of x is 4, and decreases successively by unity, while that of y increases by unity; hence the product is x+0.x3y+0.xy2+0.xy3—y1=x^—y1= product. (2) Multiply 3a2+4ax-5x by 2a2-6x+4x2. 3+4-5 2-6+4 6+8-10 -18-24-30 +12+16-20 6-10-22+46-20 ... Product=6a1—10a3x—22a2x2+46ax3—20x*. (3) Multiply 2a3-3ab2+5b3 by 2a2-5b2. Here the coefficients of a2 in the multiplicand, and a in the multiplier, are each zero; hence 2+0-3+5 2+0-5 4+0-6+10 -10- 0+15-25 4+0-16+10+15-25 Hence 4a5-16a3b+10a2b3+15ab4-25b product. = The coefficient of a being zero in the product, causes that term to dis Or, ax-(a+b) x2+(a+b+c) x3—(a+b+c)x+(a+b+c)x−(b+c) x +cx'. (7) x6-(a+d)x+(b+ad+e)x3—(c+bd+ae)x+(cd+eb)x-ce. DIVISION. 15. THE object of algebraic division is to discover one of the factors of a given product, the other factor being given; and as multiplication is divided into three cases, so, in like manner, is division. (1) When both dividend and divisor are monomials. (2) When the dividend is a polynomial, and the divisor a monomial. (3) When both dividend and divisor are polynomials. CASE I. 16. When both dividend and divisor are monomials. Write the divisor under the dividend, in the form of a fraction; cancel like quantities in both divisor and dividend, and suppress the greatest factor common to the two coefficients. 17. Powers of the same quantity are divided by subtracting the exponent of the divisor from that of the dividend, and writing the remainder as the ex Generally, amaaaa.....to m factors; a=aaa.... to n factors; bp bbbb ..... to P .... factors; bq=bbb to q factors; aaa....to n factors bbb. to q factors; = When a quantity has the same exponent in the dividend and divisor, we have Hence every quantity whose exponent is 0 is equal to 1. But we may subtract 5, the greater exponent, from 3, the less, and affect the difference with the sign a3 1 ... a hence The rule for division follows from its object, which is, having one of the factors of a product given to find the other. As in multiplication we join together the factors of a product without any sign, and without regard to order, in division we suppress from the product, i. e., the dividend, one of the factors, i. e., the divisor, to obtain the other, which is the quotient. Note.-The quotient must contain those factors of the dividend which are not in the divisor. Note, also, that dividing one of the factors of a product divides the whole product. Thus, dividing abc by a3, we divide the single factor a5, and get a2bc; so to divide 16X12 by 8, we divide 16 alone, and get 2X12 for the quotient. When there are factors in the divisor which are not in the dividend, the quotient may be expressed in the form of a fraction, as has been previously shown (2, V.). Suppressing the common factors in this case amounts to dividing both numerator and denominator by the same quantity. That such a division does not alter the value of the fraction, will be obvious from the following considerations: 1. If the numerator of a fraction be increased any number of times, the fraction itself will be increased as many times; and if the denominator be diminished any number of times, the fraction must still be increased as many times. 2. If the denominator of a fraction be increased any number of times, or the numerator diminished the same number of times, the fraction itself will, in either case, be diminished the same number of times. 3. If the numerator of a fraction be increased any number of times, the fraction is increased the same number of times; and if the denominator be increased as many times, the fraction is again diminished the same number of times, and must therefore have its original value. Hence both terms of a fraction may be multiplied by the same number, and, by similar considerations, it will appear, may be divided by the same number without changing the value of the fraction. Corollary-Rule. To multiply a fraction by a whole number, multiply the numerator of the fraction, or divide its denominator by the whole number. To divide a fraction, divide its numerator, or multiply its denominator. 1 And (x2+y°)3(x®—y?}ś=(x2+y2)-3(x2-yo)-, and so on. From this it appears that a factor may be transferred from the denominator to the numerator, and vice versa, by changing the sign of its exponent. EXAMPLES. (1) Write a b3c with the factors all in the denominator. with the factors all in one line, and also all in the denomi For more of the theory of negative exponents, see a subsequent article. 18. In multiplication, the product of two terms, having the same sign, is affected with the sign+; and the product of two terms, having different signs, is affected with the sign -; hence we may conclude, (1) That if the term of the dividend have the sign +, and that of the divisor the sign +, the resulting term of the quotient must have the sign +; because +x+ gives +. (2) That if the term of the dividend have the sign+, and that of the divisor the sign 19 the resulting term of the quotient must have the sign ; because -X — gives +. ; (3) That if the term of the dividend have the sign, and that of the divisor the sign +, the resulting term of the quotient must have the sign because + gives (4) That if the term of the dividend have the sign —, and that of the divisor the sign the resulting term of the quotient must have the sign +. like signs give +, and unlike —, the same as in multiplication. |