in either case I shall be worth only $600. The thief subtracts a positive quantity; the rogue adds a negative quantity. Again, suppose I have $1000 in my possession, but owe $400; it is immaterial to me whether a friend pays the debt of $400 or gives me $400; for in either case I shall be worth $1000. In the former case the friend subtracts a negative quantity; in the latter, he adds a positive. Or, to make the proof general: 1st. Suppose + b to be taken from the result will be and adding b to a + b we have which is, as before, equal to 2d. Suppose b to be taken from the result will be and adding +b to ab we have 3. Subtract + c from a. OPERATION. a (b+c)=a-b-c a+b b subtracted from a gives a — b; but in subtracting b we have subtracted too small a quantity by c, and therefore the remainder is too great by c, and the remainder sought is a b— c. (a - b) would be just so much too small, and the remainder sought 43. By examining the examples just given it will be seen that in every case the sign of each term of the subtrahend is changed, and that the subsequent process is precisely the same as in addition; hence, for subtrac tion in Algebra we have the following RULE. Change the sign of each term of the subtrahend from + to +, or suppose each to be changed, and then In examples 1-7, the minuend remaining the same while the subtrahend becomes in each 3 less, the remainder in each is 3 greater than in the preceding. In examples 8-14, the minuend in each becoming 3 less while the subtrahend remains the same, the remainder in each is 3 less than in the preceding. In examples 15-21, both minuend and subtrahend decreasing by 3, the remainder remains the same. 21. From 17 a x + 20 a. 18bc44xy take 25 b c 14 x y Ans. 17 ax-43bc58 xy-20 a. 22. From 384 x 74 y 18 c take 118x+74 y 27 c. Ans. 266 x 148 y + 45 c. 23. From 2 y3 + x1 — 10 x3 take x2+4y3 — x1 — 4 x3. Ans. 2x6x3-5 y3. 24. From 6 a by- 4xy + 3x z take-4aby - 3x z - 4xy. C 25. From 2+2xy + y2 take 2-2xy + y2. 26. From 2xy + y2 take-x2+2xy — y2. x2+ 44. The subtraction of a polynomial may be indicated by enclosing the polynomial in a parenthesis and prefixing the sign Thus, - taken from 2-3 may be written y3. z 2-3-2-3-(2-3 +33 — 23). (1⁄23 When a parenthesis with the sign minus before it is removed, the sign of each term within the parenthesis must be changed according to the Rule for subtraction. Thus, 23 — 23 — (203 + 33 — 23) = 2o3 — 23 — 2-3 — y3 + : A polynomial, or any number of the terms of a polynomial, can be enclosed in a parenthesis and the minus sign placed before the parenthesis without changing the value of the expression, providing the signs of all the terms are changed from plus to minus or from minus to plus. NOTE. When the sign of the first term in the parenthesis is plus, the sign need not be written. (Art. 18.) According to this principle a polynomial can be written in a variety of ways. Thus, x3-3x2y+3 x y2—y3=x3 — (3x2y — 3 x y2+y3) =x3-3x2y-(— 3xy2+y3) Remove the parenthesis, and reduce each of the following examples to its simplest form. 2. x2 - 6 a x + 2-3 — 6 x2 y — (x2 + 6 a x + x3-6 x2 y). x2 4. n2 + 2 x — (4 m2 + 3 n2 — 4c). 3 Ans. 32xy + 28c-45 y. 14 — (8 x y2 — 7 x2 y2 + 8 x3 y). -0+ 6. — (x2+7—25 x y + y3). Place in parenthesis, with the sign changing the value of the expression, 1. The last three terms of 7 x2 Ans. 7x2 14 x y 3x+4y. (14xy+3x-4y). 2. The last three terms of x2 + y2 - 3 x y + 4c. 3. The last four terms of 4 a- 7 b — 6 c — 8 d + x2. 4. The last four terms of a2 + b2 + c2 — ď2 + a3. 5. Write in as many forms as possible by enclosing two or more of the terms in parenthesis, a3 - b3 + c3 45. In subtraction, when two quantities have a common factor their difference is the difference of the coefficients of the common factor multiplied by this factor. 3. From a take bxbx2. Ans. (a - b) x2 + bx2. xs 4. From 4 x2. 6 x take a x2 + bx. x2. Ans. (4a) x— (6 + b) x. 5. From 6 a4a2-a take a3 x — a2y + az. Ans. (6x) a3 + (4 + y) a2 — (1 + z) a. 6. From ab bc take 3b+cx. 7. From a2 - bx + c√x take b x2 + cx - d√x. с 8. From xy + x2 - x2 y2 take y2 + x2y — x2 y2. |