36. Expressions may occur with more than one pair of brackets; these may be removed in succession by the preceding rules beginning with the inside pair. Thus, for example, a + {b+ (c-d)}= a + {b+c-d} = a+b+c-d, a + {b― (cd)} = a + {b-c+d} = a + b − c +d, It will be seen in these examples that, to prevent confusion between various pairs of brackets, we use brackets of different shapes; we might distinguish by using brackets of the same shape but of different sizes. A vinculum is equivalent to a bracket; see Art. 30. Thus, for example, a − [b — {c − (d − e−ƒ)}] = a−[b — {c− (d−e+ƒ)}] = a − [b — {c − d + e −ƒ}]=a−[b−c+d−e+f] a-b+c-d+e-f. In like manner more than one pair of brackets may be introduced. Thus, for example, a − b + c −d + e = a − {b − c + d− e} = a − {b − (c − d + e)}. 37. The beginner is recommended always to remove brackets in the order shewn in the preceding Article; namely, by removing first the innermost pair, next the innermost pair of all which remain, and so on. We may however vary the order; but if we remove a pair of brackets including another bracketed expression within it, we must make no change in the signs of the included expression. In fact such an included expression counts as a single term. Thus, for example, Also, a + {b + (cd)} = a + b + (c − d) = a+b+c−d, a + {b− (c−d)} = a + b − (c− d) = a + b − c +d, a — {b — (c–d)} = a − b + (c − d) = a − b + c − d. And in like manner, a-[b-{c-(d-e-ƒ)}] = a−b+ {c − (d− e−ƒ)} = a − b + c − (d − e−ƒ) = a-b+c-de-fa-b+c-d+e-f. EXAMPLES. 1. Add together 4a-5b+3c − 2d, a + b − 4c+ 5d, 2. Add together x3 + 2x2 - 3x + 1, 2x3 – 3x2 + 4x − 2, 3x2+4x2+5 and 4x3-3x2 - 5x + 9. 3. Add together x2 - 3xy + y2 + x + y − 1, 2x2 + 4xy - 3y2 - 2x - 2y + 3, 3x2 - 5xy — 4y2 + 3x + 4y − 2 and 6x2 + 10xy + 5y2 + x + y. 4. Add together x3- 2ax2 + a2x, x3 + 3ax2 and 2×3 — ax2. 5. Add together 4ab - x2, 3x2 - 2ab and 2ax + 2bx. 6. 7. From 5a-3b+4c - 7d take 2a-2b+3c — d. From x2 + 4x3- 2x2 +7x-1 take x2 + 2x3- 2x2 + 6x − 1. 8. Subtract a2. 9. ax+x2 from 3a2 - 2αx + x2. Subtract a-b-2 (cd) from 2 (a-b)-c+d. 10. Subtract (a - b) x − (b −c) y from (a + b) x + (b + c) y. 11. Remove the brackets from a-{b-(c-d)}. 12. Remove the brackets from a- · {(b − c) — d}. 13. Remove the brackets from a + 2b - 6a - {3b — (6a – 6b)}. 14. Remove the brackets from 7a-{3a - [4a – (5a – 2a)]}. 15. 16. 17. 18. 19. Also from 3a-[a + b − {a + b + c − (a + b + c + d)}]. Also from a [2b + {3c − 3a − (a + b)} + 2a − (b + 3c)]. Also from a-[5b — {a − (3c — 3b) + 2c − (a — 2b − c)}]. If a= = 2, b = 3, x= 6 and y=5, find the value of 20. Simplify a + 2x - {b + y − [a − x − (b −2y)]}. 4x3 − 2x2 + x + 1 − (3x3 — x3 − x − 7) − (x3 — 4x2 + 2x + 8). III. MULTIPLICATION. 38. We have already stated that the product of the numbers denoted by any letters may be denoted by writing those letters in succession without any sign between them; thus abcd denotes the product of the numbers denoted by a, b, c and d. We suppose the student to know from Arithmetic, that the product of any number of factors is the same in whatever order the factors may be taken; thus abc = acb= bca, and so on. 39. Suppose we have to form the product of 4a, 5b, and 3c; this product may be written at full thus, 4 xax 5 xbx 3 xc, or 4 × 5 × 3 × abc, that is 60abc. And thus we may deduce the following rule for the multiplication of simple terms: multiply together the numerical coefficients and put the letters after the product. 40. The notation adopted to represent the powers of a number, (Art. 17), will enable us to prove the following rule: the powers of a number are multiplied by adding the exponents, for a3 × a2 = a × a × a × a × a = a3 = a3+2; and similarly any other case may be established. Thus if m and n are any whole numbers, a" × a": = α m+n 41. We may if we please indicate the product of the same powers of different letters by writing the letters within brackets, and placing the index over the whole. Thus ax b2 = (ab)2; this is obvious since (ab)2 = ab × ab=a xa xbxb. Similarly, a3 × b3 × c3 = (abc)3. Thus a" × b′′ = (ab)" ; a′′ × b′′ × c′′ = (abc)"; and so on for any number of factors. The pro 42. Suppose it required to multiply a+b by c. duct of a and c is denoted by ac, and the product of b and c is denoted by bc; hence the product of a + b and c is denoted by ac+bc. For it follows, as in Arithmetic, from our notion of multiplication, that to multiply any quantity by a number we have only to multiply all the parts of that quantity by the number and add the results. Thus (a + b) c = ac + bc. 43. Suppose it required to multiply a-b by c. Here the product of a and c must be diminished by the product of b and c. Thus 44. Suppose it required to multiply a+b by c+d. It follows, as in Arithmetic, from our notions of multiplication, that if a quantity is to be multiplied by any number, we may separate the multiplier into parts the sum of which is equal to the multiplier, and take the product of the quantity by each part, and add these partial products to form the complete product. (a+b) (c+d)=(a+b) c + (a + b) d; also thus Thus (a + b) c = ac + bc, and (a+b) d = ad +bd; (a+b) (c + d) = ac + bc + ad + bd. 45. Suppose it required to multiply a-b by c+d. Here the product of a and c +d must be diminished by the product of b and c + d. Thus (a−b) (c + d) = a (c + d) − b (c + d) = ac + ad― (bc+bd) = ac + ad - bc-bd. 46. Suppose it required to multiply a+b by c-d. Here the product of a+b and c must be diminished by the product of a + b and d. Thus (a + b) (c− d) = (a + b) c − (a + b) d = ac + bc - (ad+bd) = ac + bc - ad-bd. 47. Suppose it required to multiply a-b by c-d. Here the product of a b and c must be diminished by the product of ab and d. 48. Thus (a - b) (c-d) (a - b) c − (a - b) d = ac - bc - (ad - bd) = ac - bc - ad + bd. From considering the above cases we arrive at the following rule for multiplying two binomial expressions. Multiply each term of the multiplicand by each term of the multiplier; if the terms have the same sign, prefix the sign + to their product, if they have different signs prefix the sign -; then collect these partial products to form the complete product. The rules with respect to the sign of each partial product are often enunciated thus for shortness: like signs produce +, and unlike signs produce-. 49. It appears from the preceding Articles, that corresponding to the terms - b and c which occur in two binomial factors, there is a term - bc in the product of the factors. Hence it is often stated as an independent truth that - bx c = -bc. Similarly, we observe, that corresponding to the terms - b and - c which occur in two binomial factors, there is a term be in the product of the factors; hence it is often stated as an independent truth, that -bx-cbc. These statements will be examined and explained in Chapter v. 50. The rule given in Article 48 will hold for the multiplication of any algebraical expressions. This will appear from considering a few examples. Suppose, for instance, we have to |