37. From the product of y2 - 2yz - 22 and y2+2yz – 22 take the product of y-yz-222 and y2+ yz-2z2. = 38. Find the dividend when the divisor 3a2-ab-3b2, the quotient a2b-262, the remainder = 2 ab1 –665. = The multiplication of polynomials may be indicated by inclosing each in a parenthesis and writing them one after the other. When the operations indicated are actually performed, the expression is said to be simplified. Simplify: 39. (a+b-c)(a+c-b)(b + c − a)(a+b+c). 40. (a+b) (b+c) - (c + d) (d+a) -- (a + c) (b − d). 41. (a+b+c+d)2 + (a− b − c + d)2 +(a−b+c−d)2 + (a+b− c − d)2. 42. (a+b+c)2-a (b+c− a)−b (a+c−b)−c(a+b−c). 43. (a−b) x − (b − c) a — { (b − x)(b− a)− (b −c)(b+c)}. 44. (m+n) m − { (m — n)2 — (n —m)n}. - 45. (a − b + c)2 — {a (c — a − b) — [b (a+b+c) 46. (p2+ q2)r - (p+q)(p{r — q} — q{r—p}). 47. (9 x2 y2 - 4 y1) (x2 — y2) — {3 xy - 2 y2} {3 x (x2 + y2) - 2y (y2+3xy -- x2)}y. 48. a2-{2ab-[−(a+{b− c})(a−{b-c})+2ab] − 4 bc } − (b + c)2. 49. {ac - (a - b)(b + c)} -− b { b − (a — c)}. 50. 5 {(α — b) x — cy} — 2 {a (x − y) — bx} -{3 ax (5 c-2a)y}. 51. (x − 1)(x − 2) − 3 x (x+3)+2{(x+2)(x+1) — 3} . 52. {(2a+b)2+(a − 2b)2} × { (3 a-2b)2 - (2a-36)2}. 53. 4(a−3b)(a+3b) — 2 (a− 6 b)2 — 2 (a2 + 6 b2). 54. x2 (x2+y2)2 − 2 x2 y2 (x + y)(x − y) — (x3 — y3)2. 55. 16 (a2 + b2)(a2 — b2) — (2 a − 3)(2 a +3)(4 a2+9) +(26−3)(2b+3)(462+9). 73. There are some examples in multiplication which occur so often in algebraical operations that they should. be carefully noticed and remembered. The three which follow are of great importance: From (1) we have (a+b)2 = a2+2ab+b2. That is, 74. The square of the sum of two numbers is equal to the sum of their squares + twice their product. From (2) we have (a - b)2=a2-2ab+b2. That is, 75. The square of the difference of two numbers is equal to the sum of their squares - twice their product. From (3) we have (a+b)(a−b) = a2-b2. That is, 76. The product of the sum and difference of two numbers is equal to the difference of their squares. 77. A general truth expressed by symbols is called a formula. 78. By using the double sign ±, read plus or minus, we may represent (1) and (2) by a single formula; thus, (a+b)2=a2+2ab+b2; in which expression the upper signs correspond with one another, and the lower with one another. By remembering these formulas the square of any binomial, or the product of the sum and difference of any two numbers, may be written by inspection; thus: EXERCISE XV. 1. (127)-(123)=(127+123)(127–123) =250 × 4=1000. 2. (29)=(30-1)-900-60+1=841. 3. (53)2=(50+3)=2500+300+9=2809. 4. (3x+2y)2=9x2+12xy+4y2. 5. (2a2x-5x2y)2 = 4 a1 x2 - 20 a2x23 y + 25 x1 y2. 6. (3ab2c+2a2c2)(3ab2c-2 ac2)= 9 a2b1c2-4 a*c*. 79. Also the square of a trinomial should be carefully noticed. It is evident that this result is composed of two sets of numbers: I. The squares of a, b, and c; II. Twice the products of a, b, and c taken two and two. 78. By using the double sign +, read plus or minus, we may represent (1) and (2) by a single formula; thus, (a+b)2=a2+2ab+b2; in which expression the upper signs correspond with one another, and the lower with one another. By remembering these formulas the square of any binomial, or the product of the sum and difference of any two numbers, may be written by inspection; thus: EXERCISE XV. 1. (127)2 - (123)2 = (127+123)(127 — 123) 2. (29)2(30-1)2900-60+1=841. 3. (53)=(50+3)=2500+300+9=2809. 4. (3x+2y)2=9x+12xy +4y2. 5. (2a2x-5x2y)2=4 a1x2-20 a2xy+25x+y3. 6. (3ab2c+2 ac2)(3ab2c-2 ac2)=9a2bc2-4a*c*. 7. (x + y)2= 15. (abcd)2= 8. (y-2)2= 9. (2x+1)2== 10. (2a5b)= 14. (5xy + 2)2== 16. (3 mn-4)2= 17. (12+5 ) = = |