3 times 8 away from the 7 times 8, we should obtain 4 times 8; as 56-24-32. Whence we have the general rule, if there is a negative quantity in the multiplier; the product of that quantity and the multiplicand must be subtracted from the product of the positive quantities and the multiplicand. Thus, (a+b) × (c—d). Here we see that the product of d into a+b is subtracted from the product of c into a+b; and therefore the signs of ad+bd are changed to -ad-bd. §165. By examining the answer of this last example, we shall observe a principle which will enable us to be more rapid in the multiplying operation. It is this; when we multiply a + term by a + term, the product in the answer is a + term; and when we multiply a + term by a term, the product in the final answer is a term. It will be well for the pupil to explain this. By understanding this principle, we are able to set the final answer down at first; as we have only time to remember that, §166. When the signs are alike, the product is +. §167. When the signs are unlike, the product is. 47. Multiply x+8% by 3x2-7xz. Ans. 3x+17x3z-56xz2. 48. Multiply x+xy+y' by x-y. 49. Multiply 2x+3y by 3x-4y. Ans. x3-y3. Ans. 6x+xy-12y'. 50. Multiply b+b2x+bx2+x3 by b-x. 51. Multiply a-b by c-d. Ans. b4--x4. Here we see that in subtracting d times a-b, we change the signs of ad-bd to ―ad+bd. §168. Whence we learn, + multiplied by +, produces + + multiplied by, produces -multiplied by, produces +. And by remembering this we can always set down the 54. Multiply 2a- 5y by a-2y. Ans. 2a-9ay+10y. 55. Multiply a2+ac-c3 by a- c. Ans. a3-2ac2+c3. 56. Multiply a+b-d by a-b. 57. Multiply 4x-5a-26 by 3x-2a+5b. Ans. 12x-23ax+14bx+10a2-21ab-106. 58. Multiply y3 3 by x-y-z. 59. Multiply 6+xy-a-my3 by a3-3x2+y*. 60. Multiply 2ab3ac2+4b3c2 — 1 §169. In order to facilitate the practice of multiplication, it is best to observe the following method. First, determine the sign, then the co-efficient, then the letters in their order, and then the exponents. General Properties of Numbers. $170. We have before stated that algebraical operations, §137, by reason of the quantities themselves being retained in their original value, do show us, in their results, important general principles. We will here make a few multiplications of some quantities, whose results show us some remarkable general properties of numbers. These properties the pupil should remember, as they are of frequent use in the subsequent parts of this study. §171. Suppose we have two numbers, a and b, of which a is the greatest. Then their sum = a+b; and their difference =α b. Then a+b a-b a2+ab —ab-b3 By this operation, we find the general property of numbers which it would be difficult to find by any arithmetical operation. It is that, if we multiply the sum of two numbers by their difference, the product will be the difference of the squares of those numbers. §172. Again, take the same quantities, and multiply their sum, by their sum. a+b a+b a2+ab +ab+b a2+2ab+b2 By this operation we find the following general property. The square of the sum of two numbers is equal to the square of the first number, plus twice the product of the two numbers, plus the square of the last number. §173. Again, take the same quantities, and multiply their difference, by their difference. Therefore, the square of the difference of two numbers, is equal to the square of the first number, minus twice the product of the two numbers, plus the square of the second. §174. The only difference between the square of the sum, and the square of the difference, is in the second term ; being in one, positive, and in the other, negative. Let us subtract one from the other. a2+2ab+b2 a2-2ab+b2 4ab The actual difference between the square of the sum, and the square of the difference, is four times the product of the two numbers. §175. If when the sum of two quantities has been raised to the second power and the co-efficient of the second term has been rejected, the quantity thus obtained be multiplied by the difference of the two original quantities; the result will be the difference of the third powers of the two quantities. Also, if we perform the same operation with the difference of the two quantities, multiplying by their sum, we shall obtain the sum of the third powers of the two quantities. Thus, $176. It is often the case that it is better to denote the multiplication of compound quantities, than to perform it; on account of operations that follow. Thus, a-b times x+y, may be written (ab). (x+y). No general rule can be given to determine when one method is preferable to the other. Experience is the best teacher in this partícular. But it was thought best to mention it in this place. When, after the multiplication has been denoted, the several terms are actually multiplied, the expression is said to be expanded. |