(4) x2+x+1 and x2+x-1; a+2b+3c and a+2b-3c. (5) a+2b+3c and a-2b+3c; a+2b+3c and a-2b-3c. (6) a2 — 2ab+b2 and a2+2ab+b2; ‡àa—ƒœ3+‡æa and ‡æa+}∞2+‡x2. - 1. (x+2). (x+3)= x2+6+(2+3)x, by (D), = x2+6+5x, or x2+5x+6. 2. (5-2x). (7-2x)=35+ (2x)-(7+5)2x, by (E), =35+4x2-24x, or 35-24x+4x2. 3. (2+x). (3−x)=6−x2+(3−2)x, by (F), =6-x2+x, or 6+x-x2. 4. (2y-x).(3y+x2)=6y3-x2+(2y-3y)x", by (G), However, if the student have difficulty in remembering the formulæ D, E, F, G, he can always in simplifying expressions proceed as follows: 5. (2y-x). (3y+x)=(2y-x2)3y+(2y-x2x2 6. (a2-ab+b2). (a−b) = (a2 — ab+b2)a— (a2 — ab+b2)b =a3-ab+ab3-(ab-ab3+b3) =a3— a2b+ab2 — a2b+ab3 — b3 - 7. (a2+b2+c2). (x2 + y2 — z3) = (a2+b23 +c3)x® + (a2+b2 + c3)y3 — (a2 + b2 + c2)≈a =a2x2+b2x2+c2x2+a2y3+b2y2+c2y3 — a2x2 — b2x2- cz3. Examples for practice. Ex. XII. D. E. F. G. Write down the product of 1. x+2 and +4; x-6 and x-11; 5-≈ and 2+x. 2. 2ax-1 and 2ax-3; u+4b and a-2b; x+a and x+c. 3. x2+3y and 2-2y; aTM+bTM and aTM-2bm; a-b3 and 2aa — 6b3. 4. x2-xy+y2 and x+y; x2+3x-2 and x+3; a2+3ab+4b2 and 2ab-b2. 5. (a3+2a2b+2ab2+b3) and (a3— 2a2b+2ab2 — b3); a+b+c+d multiplied 1st by a-b-c+d, 2nd by a-b-c-d; (x2-x+1). (x2+x+1) and Simplifications worked out. 31. 1. Simplify (a+b)2—2b (a+b)+2b3. (a+b)2-2b (a+b)+262=a2+2ab+b2-(2ab+2b3)+262 =a2+2ab+b2-2ab-2b2+262 =a2+b2. 2. Write down the product of (x+1). (x-2). (x+3). =(x2-x-2)x+(x2-x-2)3 =x3+2x2-5x−6. 3. Simplify (a+b). (b+c)+(a+c). (d—b)− (a+d). (c+d). (a+b). (b+c)+(a+c). (d—b)− (a+d). (c+d) =(a+b)b+(a+b) c+(a+c) d−(a+c) b-(a+d) c-(a+d) d =(ab+b2)+(ac+bc)+(ad+cd)-(ab+cb)-(ac+de) - (ad+d3) =ab+b2+ac+be+ad+cd-ab-cb-ac-dc-ad-d2 = b2-d3. | 4. Simplify (22 — 9x+20). (x2 — 5x) – (x2-13x+42). (x2 — 6x). (x2-9x+20). (x2-5x)-(x2-13x+42). (x2-6x) = (x^— 9×3+20x2-5x3+45x2-100x) — (x1—13x3+42x2-6x3+78x2-252x) = x2-9x3+20x2-5x2+45x2-100x-x2+13x3-42x2 +6x-78x2+252x =5x3-55x2+152x. Ex. XIII. Simplify the following expressions: (1) (x-1)(x+2)-(x−2)(x+1). (2) (a−b) (a−c)+(b−c) (b−a). (3) 2x (x+y)(x+ z) − (x + z) ( x −z) 2y. (4) (2−x2)(2+x2)+(5x2+2)(1 − 7x2)+9x2 (2x+1)(2x−1)− 6. (5) (3b+2a) (3b−2a)+(3a−b) (3a+b) −2 (b−a) (b+a). (6) (n+1)(n+2) (n+3)-(n+1)(n+2)-n3. (7) (a+b)-(c-d)2 + (a−b)2 + (c+d)3. (8) (x-y)2+(x+y)2+2 {(x − y) (x+y)+(x+y) (x−y)}. (9) a2 (3a-5b)-(2a−b)3 (a−2b) + (a2+ab+b2)(a−b)—b3, and find its numerical value when a=2, b=}. (10) 2x (x+4) (x+3)-(x−1)2 (2x-1)-(19x2-20x+1). (11) (a2+ab+b2) (a+b) — (a2 — ab+b3) (a−b). (12) 3 (a+2x)2−2 (a+2x) (a−2x)+(a−2x)3. (13) (x2+y+23)2 — (x2+ y2+≈3). (14) (a+b+c)+(a+b−c)2+(b+c-a)2+(c-a+b). (15) (a+b+c)2-{a (b+c− a)+b (a+c—b)+c (a+b−c)}. (16) (a+b+c)2− (a−b+c)2+(a+b−c)2 - (b+c-a)3, find its numerical value when a=b=c=-4. (17) (2x+3y+4)3-{2x (3y+4-2x)+3y (2x+4-3y) +4(2x+3y-4)}. (18) (a"-b"-c") (a+b+c), find its numerical value when a=4, b=2, c=1, n=2. (19) {(x-1) (x-2)+3} {(x−2) (x-3)+4}+8x3+55x+175. (20) (2a+3b) (3+4c)-(4c+b) (6+2a)-(2a+4c) (3b-6). (21) {(x+y)2+(x − y)2} {(x + y)2 — (x-y), find its numerical value when x=y=. (22) (x+a) (x+b) (a−2x)-a (b+x) (a−x)+2x2 (2x-b). (24) (1−ax)(1 − bx) (1 − cx) − (1 − cx)2+(1— ax) (bx−1). (25) (x+1)(x+2) (3−x) (4−x) +x (x-3) (x+4)(x+3) -2(x-1) (5+x) (x+1) (x−6). (26) Shew that (x-2y) 2xy-xz (x−≈)+(2y−≈) 2yz =(x-2y) (x-2) (2y-x). (27) Shew that (a2+b2+c2) (x2 + y2 + z2) − (ax+by+cz)2 = (ay—bx)2 + (bz — cy)2 + (cx — az)2. (28) Shew that (x-x-y)(x+y)-2xyz(x+y+2) — x(x2 — y2+x2) (y + z) = (xy+yz) (y2 — x2 + z2). (29) Shew that a(b+c)+b(a+c)2 + c(a+b) - {(a+b) (a-c) (b-c) +(a-b) (ac) (b+c)-(a−b) (a+c) (b-c)}=12abc. 2 (30) Shew that (+1)+(+9)2 + (+) * a (u 台) 32. CASE 1. When the dividend and divisor are both simple quantities. RULE. "Write the divisor under the dividend in the form of a fraction, and then divide the two terms of the fraction by all the factors which are common to both; if the two quantities have the same sign, prefix to the quotient thus obtained the sign+, if they have different signs, the sign-." Ex. 2. Divide 6abc by -2a; and -10xyz by -5y. Ex. 3. Divide -14abc by -7x'c, and 20m3n p2 by -6m2q2. - 14a2bc 2a2b 20m3n3p2 10n2p2 ; 33. The rule of signs in Division follows immediately from the rule of signs in Multiplication. 34. CASE 2. When the dividend is a compound quantity, and the divisor a simple one. RULE. "Divide each term of the dividend by the divisor, as in Case 1." |