Thus the sum of +7 and −5 is +2, which is numerically less than either 7 or 5. So, also, the sum of +a and —b is a-b. In this case the algebraic sum is numerically the difference of the two quantities. This is one instance among many in which the same terms are used in a much more general sense in the higher mathematics than they are in Arithmetic. 44. When dissimilar terms have a common literal part, we may regard the other factors as the coefficient of the common letter or letters. The sum of the terms will then be expressed by inclosing the sum of the coefficients in a parenthesis, and prefixing it to the common letter or letters. cx2 Thus the sum of ax2, bx2, and cx2 may be written (a+b+c)x2. EXAMPLES. 1. Add ax, 2bx, and 3mx. 2. Add 3axy2, 2bxy2, and -5axy2. Ans. (a+2b+3m)x. Ans. (2b—2a)xy2. Ans. (7a+1)x-2y. 3. Add 2ax+3y, 5ax-y, and x—4y. 5. Add mx+ny, 3ax-2y, and 4bx+ay. 6. Add 4m √x+3, 2a√x−1, and b√x+y. 7. Add 3ax2+2bx-1, 4bx2-ax+3, and mx2—nx+5. 8. Add 2ax2+3bx3 −7, 3mx1—nx3+2, and 4xa — ax3 +1. 9. Add amx3+bnx2+cx, bmx3 — anx2+ax, and cmx3-nx2 +3bx. 10. Add (a-b) V and (a+b-c) √x. CHAPTER III. SUBTRACTION. 45. Subtraction is the operation of finding the difference be tween two quantities or sets of quantities. The quantity to be subtracted is called the subtrahend; the quantity from which it is to be subtracted is called the minuend; the quantity which is left after the subtraction is called the remainder. Let it be required to subtract 8-3 from 15. Now 8-3 is equal to 5; and 5 subtracted from 15 leaves 10. The result, then, must be 10. But, to perform the operation on the numbers as they were given, we first subtract 8 from 15, and obtain 7. This result is too small by 3, because the number 8 is larger by 3 than the number which was required to be subtracted. Therefore, in order to correct this result, the 3 must be added, and the operation may be expressed thus, 15-8+3=10. Again, let it be required to subtract c-d from a-b. It is plain that, if the part c were alone to be subtracted, the remainder would be a-b-c. But, since the quantity actually proposed to be subtracted is less than c by d, too much has been taken away by d, and therefore the true remainder will be greater than a-b-c by d, and may hence be expressed thus, a-b-c+d, where the signs of the last two terms are both contrary to what they were given in the subtrahend. Hence we perceive that a quantity is subtracted by simply changing its sign. In practice it is most convenient to write the quantities so that similar terms may be found in the same column. B 46. Hence we deduce the following RULE. Write the subtrahend under the minuend, arranging similar terms in the same column. Conceive the signs of all the terms of the subtrahend to be changed from + to, or from - to, and then proceed as in addition. 9. From 3a+5b-2c subtract 2a-b. Ans. a+6b-2c. 10. From 5abc-2b-6 subtract 3abc-2b+1. 11. From 4a2-7a+3x subtract a2+3a-2x. Ans. 2abc-7. Ans. 3a2-10a +5x. 12. From a-b+2m-x subtract 3x+m-4b+a. 13. From 2x3-x2y+5xy2 subtract x3-2xy2+y3. 14. From m+n subtract m—n. 15. From m+n+x subtract -MN-X. 16. From 5a2-3a-7 subtract -2a2-4a+10. 17. From m1+3m3-4m2-2m+1 subtract m2 - 2m3 + m2 -3m+5. 18. From 2-5x+10x3-3 subtract x+5x1-10x3+3. 19. From 3a2+ax+2x2-14a2x subtract x2 - 15a2x+2a2 -4ax. 20. From 6abx-4mn+5ax subtract 3mn+6ax-3abx. Subtraction may be proved, as in Arithmetic, by adding the remainder to the subtrahend. The sum should be equal to the minuend. 47. It will be perceived that the term subtraction is used in a more general sense in Algebra than in Arithmetic. In Arithmetic, where all quantities are regarded as positive, a number is always diminished by subtraction. But in Algebra the difference between two quantities may be numerically greater than either. Thus the difference between + a and b is a+b. The distinction between positive and negative quantities may be illustrated by the scale of a thermometer. The degrees above zero are considered positive, and those below zero negative. From five degrees above zero to five degrees below zero, the numbers stand thus, +5, +4, +3, +2, +1, 0, −1, −2, -3, -4, -5. The difference between a temperature five degrees above .zero and one which is five degrees below zero, is ten degrees, which is numerically the sum of the two quantities. Ten is said to be the algebraic difference between +5 and -5. 48. When dissimilar terms have a common literal part, the difference of the terms may be expressed, as in Art. 44, by inclosing the difference of the coefficients in a parenthesis, and prefixing it to the common letter or letters. Thus the difference between ax2 and bx2 may be written (a—b)x2. EXAMPLES. 1. From ax2y2 subtract -3x2y2. Ans. (a+3)x2y2. Ans. (2a-5b)x+4y. Ans. (m-3a)x+(n+2)y. 4. From 4m √x+3 subtract 2a √x-1. 5. From 2ax+3bx3-7 subtract 3ma-nx3+2, 6. From amx3+bnx2 + cx subtract bmx3-anx2+ax. 7. From m+am+bm subtract am+bm+cm. 8. From 1+3аx2+5a2x2+7a3x subtract x2-3ax3-5a2x*. 49. Use of the Parenthesis.-If we wish to indicate that one polynomial is to be subtracted from another, we may inclose it in a parenthesis, and prefix the sign minus. Thus the expression a—b—(m—n+x) indicates that the polynomial m-n+x is to be subtracted from the polynomial a-b. Performing the operation indicated, we have a-b-m+n-x. indicates that the polynomial m-n+x is to be added to the polynomial a—b, and the result is a−b+m—n+x. Hence we see that a parenthesis preceded by the plus sign may be removed without changing the signs of the inclosed terms; and, conversely, any number of terms, with their proper signs, may be inclosed in a parenthesis, and the plus sign written before the whole. But if the parenthesis is preceded by the minus sign, the signs of all the inclosed terms must be changed when the parenthesis is removed; and, conversely, any number of terms may be inclosed in a parenthesis, and preceded by the minus sign, provided the signs of all the inclosed terms are changed. 50. According to the preceding principle, polynomials may be written in a variety of forms. These expressions are all equivalent, the first form being the simplest. EXAMPLES. Reduce the following expressions to their simplest forms. 1. 2a3-5a2b+3ab2— (a3+b3 —ab2). Ans. a3-5a2b+4ab2-b3. |