SUBTRACTION. SET down in one line the first quantities from which the subtraction is to be made; and underneath them place all the other quantities composing the subtrahend: ranging the like quantities under each other, as in Addition. Then change all the signs (+ and -) of the lower line, or conceive them to be changed; after which, collect all the terms together as in the cases of Addition*. Take-2xy +2 7√√xy + 3—2xy 9x2-12 +5b+x2 Rem. This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs and, by which they are expressed and represented. So that, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive cne from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity. In like manner, to subtract a negative quantity, is the same in effect as to add or unite an equal positive one. So that, by changing the sign of a quantity from to, or from- to +, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted, by barely changing its sign. From 2x 4a-265 take 8 56 + a + bx.. From 4a From 8a 12.x, take 4a From 3a + b + c From 3a + b + c - d 10, take c + 2a d. 19+ 3a. -c+ b2. abc, take b2 +ab2 9a abc. From 12x+6a-4b+ 40, take 46 3a + 4r+6d-10. -- 1041 MULTIPLICATION. This consists of several cases, according as the factors are simple or compound quantities. CASE 1. When both the Factors are Simple Quantities : FIRST multiply the co-efficients of the two terms together, then to the product annex all the letters in those terms, which will give the whole product required. Note*. Like signs, in the factors, produce + and unlike signs, in the products. EXAMPLES. That this rule for the signs is true, may be thus shown. 1. When a is to be multiplied by +c; the meaning is, that a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that+ax+c makes+ac. 2. When SUBTRACTION. SET down in one line the first quantities from which the subtraction is to be made; and underneath them place all the other quantities composing the subtrahend: ranging the like quantities under each other, as in Addition. Then change all the signs (+ and -) of the lower line, or conceive them to be changed; after which, collect all the terms together as in the cases of Addition*. *This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs and —, by which they are expressed and represented. So that, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive ene from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity. In like manner, to subtract a negative quantity, is the same in effect as to add or unite an equal positive one. So that, by changing the sign of a quantity from to, or from — to +, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted, by barely changing its sign. This consists of several cases, according as the factors are simple or compound quantities. CASE 1. When both the Factors are Simple Quantities : FIRST multiply the co-efficients of the two terms together, then to the product annex all the letters in those terms, which will give the whole product required. Note*. Like signs, in the factors, produce + and unlike aigns, in the products. EXAMPLES. That this rule for the signs is true, may be thus shown. 1. When a is to be multiplied by + c; the meaning is, that a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that +ax+c makes+ac. 2. When Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them, begin at the left-hand side, and multiply from the left hand towards the right, in the manner that we write, which is contrary to the way of multiplying numbers. But in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lines above, when there are such like quantities; which is the easiest way for adding them up together. In many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum with a sign of multiplication between them. As (a + b) × (a — b) × 3ab, or a+b.a-b. 3ab. EXAMPLES FOR PRACTICE. 1. Multiply 10ac by 2a. 2. Multiply 3a2 2b by 3b. 3. Multiply 3a + 2b by 3a 26. Ans 20a2 C. Ans. 9a2b-6b2. Ans. 9a2-462. Ans. a4b4. 5. Multiply a3 + a2b+ab2 + b3 by a-b. 6. Multiply a2 + ab + b2 by a2 ab + b2. 7. Multiply 3x2 -2xy + 5 by x2 + 2xy-6. 8. Multiply 3a3-2ax + 5x2 by 3a2 9. Multiply 3x3 + 2x2 y2 + 3y33 by 2x3-3x3 y2 + 3y3. 10. Multiply a + ab + b2 by a- 2b. 4ax 7x2. DIVISION. DIVISION in Algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left-hand side, and dividing all the parts of the dividend by the divisor, when they can be se divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases. CASE |