expressed thus, aaXa, or aaa, which, for the sake of brevity, is written a3. Hence, the exponent of a letter in the product, is equal to the sum of its exponents in the two factors. rule of the exponents. 3. What is the product of a2 by a2? . . 4. What is the product of a2b by ab? This is termed, the Ans. aaaa, or a*. Ans. aaabb, or a3b3. 5. What is the product of 2ab2 by 3ab? Ans. 6aabbb, or 6a2b3. Hence, the RULE, FOR MULTIPLYING ONE POSITIVE MONOMIAL BY ANOTHER. pro Multiply the coefficients of the two terms together, and to their duct annex all the letters in both quantities, giving to each letter an exponent equal to the sum of its exponents in the two factors. NOTE. It is customary to write the letters in the order of the alphabet. Thus, abXc is generally written abc. Ans. 20a2b2x3y. Ans. 15a+b+c*. 14. What is the product of 3abc by 5ab2c3? 15. What is the product of 7xyz by 8x3yz? . . Ans. 56x1y3z2. NOTE. The learner must be careful to distinguish between the coëfficient and the exponent. Thus, 2a is different from a3. To fix this in his mind, let him answer such questions as the following: ART. 70.—1. Suppose you purchase 5 oranges at 4 cents a piece, and pay for them, and then purchase 2 lemons at the same price; what will be the cost of the whole? 5 oranges, at 4 cents each, will cost 20 cents; 2 lemons, at 4 cents each, will cost 8 cents, and the cost of the whole will be 20+8=28 cents. The work may be written thus: 5+2 4 20+8=28 cents. If you purchase a oranges at c cents a piece, and b lemons at c cents a piece, what will be the cost of the whole? The cost of a oranges, at c cents each, will be ac cents; the cost of b lemons, at c cents each, will be be cents, and the whole cost will be ac+bc cents. The work may be written thus: a+b с ac+bc Hence, when the sign of each term is positive, we have the following RULE, FOR MULTIPLYING A POLYNOMIAL BY A MONOMIAL. Multiply each term of the multiplicand by the multiplier. EXAMPLES. 2. Multiply a+d by b. 3. Multiply ac+be by d.. 4. Multiply 4x+5y by 3a. 5. Multiply 2x+3z by 2b. 6. Multiply m+2n by 3n. 7. Multiply x+y by ax. 8. Multiply a+y2 by xy.. 9. Multiply 2x+5y by abx. 10. Multiply 3x2+2xz by 2xz. 11. Multiply 3a+2b+5c by 4d.. 12. Multiply be+af+mx by 3ax. 13. Multiply ab+ax+xy by abxy.. . ART. 71.-1. Let it be required to find the product of x+y by a+b. Here the multiplicand is to be taken as many times as there are units in a+b, and the whole product will evidently be equal to the sum of the two partial products. Thus, x+y a+b ax+ay the multiplicand taken a times. bx+by the multiplicand taken 6 times. ax+ay+bx+by the multiplicand taken (a+b) times. If x=5, y=6, a=2, and b=3, the multiplication may be arranged thus: 5+6 2+3 10+12=the multiplicand taken 2 times. 15+18 the multiplicand taken 3 times. Hence, when all the terms in each are positive, we have the following ᎡᏌᏞᎬ, FOR MULTIPLYING ONE POLYNOMIAL BY ANOTHER. Multiply each term of the multiplicand by each term of the mulli plier, and add the products together. 9. Multiply 4x+5y by 2a+3x. Ans. 8ax+10ay+12x2+15ay. 10. Multiply 3x+2y by 2x+3y.. 11. Multiply a2+b2 by a+b, 12. Multiply 3a2+262 by 2a2+362. 13. Multiply a2+ab+b2 by a+b. 14. Multiply c3+d3 by c-+d.. Ans. 6x+13xy+6y'. Ans. a+ab+ab2+b3. Ans. 6a++13a2b2+6b1. Ans. a3+2ab+2ab2+b3. Ans. c+cd+c3d+d*. 15. Multiply 2+2xy+y2 by x+y... Ans. x3+3x2y+3xy2+y3. SIGNS. ART. 72. In the preceding examples, it was assumed that the product of two positive quantities, is also positive. It may, how ever, be shown as follows: 1st. Let it be required to find the product of +b by a. The quantity b, taken once, is +b; taken twice, is evidently, +26; taken 3 times, is +36, and so on. Therefore, taken a times, it is ab. Hence, the product of two positive quantities is positive; or, as it may be more briefly expressed, plus multiplied by pluo, gives plus. 2d. Let it be required to find the product of -b by a. REVIEW. To what is the exponent of a letter in the product equal? What is the rule for multiplying one positive monomial by another! 70. What is the product of a plus b, by c? When all the terms in each are positive, how do you multiply a polynomial by a monomial? 71. When all the terms in each are positive, how do you find the product of two poly. nomials? The quantity -b, taken once, is -b; taken twice, is -2b; taken 3 times, is -36; and hence, taken a times, is ab; that is, a negative quantity multiplied by a positive quantity, gives a negative product. This is generally expressed, by saying, that minus multiplied by plus, gives minus. 3d. Let it be required to multiply b by -a. Since, when two quantities are to be multiplied together, either may be made the multiplier (Art. 67), this is the same as to multiply -a by b, which gives-ab. That is, a positive quantity multiplied by a negative quantity, gives a negative product; or, more briefly, plus multiplied by minus, gives minus. 4th. Let it be required to multiply -3 by -2. The negative multiplier signifies, that the multiplicand is to be taken positively, as many times as there are units in the multiplier, and then subtracted. The product of -3 by +2 is -6, then, changing the sign to subtract, the 6 becomes +6; and, in the same manner, the product of —b by -a is +ab. Hence, the product of two negative quantities is positive; or, more briefly, minus multiplied by minus, gives plus. NOTE.-The following proof of the last principle, that the product of two negative quantities is positive, is generally regarded by mathematicians as more satisfactory than the preceding, though it is not quite so simple. The instructor can use either method. 5th. To find the product of two negative quantities. To do this, let us find the product of c-d by a—b. Here it is required to take c―d as many times as there are units in a—b. It is obvious that this will be done by taking c-d as many times as there are units in a, and then subtracting from this product, c―d taken as many times as there are units in b. Since plus multiplied by plus gives plus, and minus multiplied by plus gives minus, the product of c-d by a, is ac-ad. In the same manner, the product of c-d by b, is bc-bd; changing the signs of the last product to subtract it, it becomes-bcbd; hence the product of -d by a-b, is ac-ad-bcbd. But the last term, +bd, is the product of -d by -b, hence the product of two negative quantities is positive; or, more briefly, minus multiplied by minus produces plus. The multiplication of c-d by a-b may be written thus: c-d ac-ad-c-d taken a times. -be+bd-c-d taken b times, and then subtracted. ao-ad-bc+bd The operation may be illustrated by figures; thus, let it be required to find the product of 7-4 by 5—3. 7-4 5-3 35-20 -21+12 35-41+12 We first take 5 times 7-4; this gives a product too great, by 3 times 7-4, or 21-12, which, being subtracted from the first product, gives for the true result, 35-41+12, which reduces to +6. This is evidently correct, for 7-4 =3, and 5—3—2, and the product of 3 by 2 is 6. From the preceding illustrations, we derive the following GENERAL RULE, FOR THE SIGNS. Plus multiplied by plus, or minus multiplied by minus, gives plus. Plus multiplied by minus, or minus multiplied by plus, gives minus. Or, the product of like signs gives plus, and of unlike signs gives minus. From all the preceding, we derive the GENERAL RULE, FOR THE MULTIPLICATION OF ALGEBRAIC QUANTITIES. Multiply every term of the multiplicand, by each term of the multiplier. Observing, 1st. That the coefficient of any term is equal to the product of the coefficients of its factors. 2d. That the exponent of any letter in the product is equal to the sum of its exponents in the two factors. 3d. That the product of like signs, gives plus in the product, and unlike signs, gives minus. Then, add the several partial products together. Multiply 9-5 by 8-2. Ans. 40-15-25=5X5. Ans. 80-52-28=7X4. Ans. 143-181+56=18=6X3. Ans. 30-41-15=-26-13X-2. Ans. 72-58+10=24-4X6. 6. Multiply 8-7 by 5-3. . . . Ans. 40-59+21=2=1X2. REVIEW.-72. What is the product of +b by +a? Why? What is the product of-b by a? Why? What is the product of +b by-a? Why? What is the product of -3 by -2? What does a negative multiplier signify? What does minus multiplied by minus produce? What is the gen. eral rule for the signs? What is the general rule for the multiplication of algebraic quantities? |