but this product is evidently too large, as we are not required to multiply by the whole value of x, but only by its excess over the value of y. We must, therefore, multiply by y, and subtract the product, a y + from the former; but when subtracted, it becomes b y, In this operation, + multiplied by -, gives the product. 12. Multiply 3 a + by by df. 13. Multiply a b + c + x by 2 c — d. 8 x. 19. Multiply a + 5 + x by a -x-5. 20. Multiply h m + 16 + y by 9 b — 8 b. ANS. α x b x a y + by. in We first multiply a— b by x, and the product is а х .b x. But we are not required to multiply ab by the whole value of x, but only by its excess over the value of y; we must, therefore, multiply by y, and subtract this product from the former. The product of ab multiplied by y, is a yby, and when subtracted, it becomes-ay+by. Let a = 15, b = 5, x = 12, and y = 8; then we -ay+by a x − b x — a y+by. Hence it that appears, 120+40 60-120+40=40. multiplied by - gives + in the product. This is a principle which is very apt to perplex beginners. To use a common expression, they cannot understand how the multiplying of less than nothing by less than nothing, can give a real or positive quantity. Thus stated, the subject is, indeed, quite nexplicable; but the whole difficulty vanishes when it is remembered that there are, in fact, two operations carried on at the same time, namely, Multiplication and Subtraction. We first multiply by the negative quantity, as if it were positive; and then, by changing the signs of all the terms, subtract the product from the quantity already obtained. 22. Multiply bac— bc by f―c. 23. Multiply a-ab-4 by b-5. 24. Multiply 6 a by-6 a. 25. Multiply 12 a b 8 by a - 12. 26. Multiply 8 by — 6. 27. Multiply 8 a b-3 x by 2 a b — 5 x SECTION IV. General Rule for Multiplication. In the preceding sections of this Chapter, have been developed all the rules to be observed in the multiplication of algebraic quantities. To facilitate practice, they will now be repeated together. I. MULTIPLICATION. Multiply each term of the multiplicand by each term of the multiplier. 2. SIGNS. When both terms have the same sign, the product has the sign +; but when they have different signs, the product has the sign 3. COEFFICIENTS. Multiply the coefficients of both terms together, and use their product. 4. LETTERS. Write the letters of both terms in order, one after the other. 5. REDUCTION. Add together the several products by their proper signs, and unite such as are similar into one term. 1. Multiply x + 2 x y + y by x — y. ANS. xx+2x xy-2xyyyy. 2. Multiply x + xy + y by x - x y + y. 3. Multiply 3x-2xy + 5 by x + 2 x y — 3. 4. Multiply 2 a 3 ax+4x by 5 a-6 ax-2x. 5. Multiply +36-6 by 4 x8b8. b 6. What is the product of 77-2 m 9 multiplied by 3 - 11 m? 7. Multiply a + b + 6 by a − b − 6. 8. Multiply a bbc + c d by a bbc - cd. 9. What is the product of 5 + b c multiplied 5+ 10. What is the product of 7 a-3x+5 multi. plied by 4 a + d? 11. Multiply b b b c + c c by b + c. 12. Required the product of 3 mm x-m x + x multiplied by 2 my. 13. What is the product of x x + xy + 3 z multiplied by x + 1 ? 14. Multiply a+b-c by 1-x. 15. What is the product of a a multiplied by a + x? - 4 α x + x x 16. Required the product of a a-2 ay+yy multiplied by a—y. 17. Multiply a + c g by 4 aa+3 cg. 18. What is the product of m-n+ z multiplied by 5 m n 2yz? 19. Required the product of 16 am +3 ac multiplied by a y +4 m. 20. Multiply a a-bb8 by x-yyy. 21. What is the product of xxyyy-7 multi plied by xy? 22. Required the product of a+b+c+d mul tiplied by a - - b с d. 12 x by y — 1. 2b 24. Multiply 3 a +5-26 by 5 a + 2 b — 5. CHAPTER V. DIVISION. SECTION I. Simple Quantities. 1. Ir you divide 15 cents equally among 3 boys, how many cents will each boy have? ANS. 5 cents. When any given quantity is to be separated into a certain number of equal parts, the value of one of those parts is determined by Division. Thus, if we were to count off 15 cents into three equal piles, we should find that each of those piles would contain 5 cents; that is, 3 is contained 5 times in 15. The quantity to be divided is called the Dividend; the quantity, denoting the number of equal parts into which the dividend is to be divided, is called the Divisor; and the value of one of those parts is called the Quotient. Thus, in the above example, 15 is the Dividend, 3 is the Divisor, 5 is the Quotient. As Division is the reverse of Multiplication, the divisor and quotient being multiplied together, will |