57. The process in the multiplication of factors, one or both of which contains more than two terms, is similar to the processes which we have been describing, as may be seen from the following examples: (3) Multiply 3x2+4xy-ya by 3x2-4xy+y2. (4) To find the continued product of x+3, x+4, and x+6. To effect this we must multiply x+3 by x+4, and then multiply the result by +6. x+ 3 x + 4 x2 + 3x + 4x+12 x2+7x+12 x + 6 x2+7x2+12x +6x2+42x+72 x+13x2+54x+72 Note. The numbers 13 and 54 are called the coefficients of 2 and x in the expression + 13x2 + 54x + 72, in accordance with Art. 44. (5) Find the continued product of x+a, x+b, and x+c. x+a x+b x2+ax + bx+ab x2+ax+bx+ab x + c x3+ax2+bx2 + abx + cx2+acx+bcx + abc x3+(a+b+c)x2 + (ab + ac+bc) x+abc Note. The coefficients of 2 and x in the expression just obtained are a+b+c and ab+ac+bc respectively. When a coefficient is expressed in letters, as in this example, it is called a literal coefficient. EXAMPLES.-VIII. Multiply 1. x+3 by x+9. 2. 4. x-8 by x-7. 5. +15 by x- 3. x-12 by x+10 6. y-6 by y+13. 8. x2-6x+9 by x2-6x+5. 10. a3-3a+2 by a3-3a2+2. 12. x2 + xy + y2 by x2—xy+y2. 14. a2-x2 by aa+a2x2+x1. 15. x3-3x2+3x-1 by x2+3x+1. 16. x3+3x2y+9xy2+27y3 by x-3y. 17. a3+2ab+4ab2+8b3 by a-2b. 18. 8a3+4a2b+2ab2+b3 by 2a-b. 7. 2-4 by x2+5. 9. x2+5x-3 by x2-5x-3. 11. x2-x+1 by x2+x−1. 13. x2+xy+y2 by x-y. 19. a3-2ab+3ab2+4b3 by a2-2ab-3b2. Find the continued product of the following expressions: 27. x-α, x+α, x3+a2, x2+a3. 28. x-a, x+b, x−c. 29. 1-x, 1+x, 1+x3, 1+x^. 30. x−y, x+y, x3 — xy +y3, x2 + xy + y2. 31. a-x, a+x, a2+x2, aa+x^, a3 +x3. Find the coefficient of x in the following expansions : 32. (x-5) (x-6) (x+7). 33. (x+8)(x+3)(x-2). 34. (x-2)(x-3)(x+4). 35. (x-a) (x-b) (x−c). 36. (x2+3x-2) (x2 - 3x+2) (x-5). 37. (x2-x+1)(x2 + x − 1) (x2 — x2 + 1). 38. (x2-mx+1) (x2 — mx − 1) (x1 — m2x − 1). 58. Our proof of the Rule of Signs in Art. 55 is founded on the supposition that a is greater than b and c is greater than d. To include cases in which the multiplier is an isolated negative quantity we must extend our definition of Multiplication. For the definition given in Art. 36 does not cover this case, since we cannot say that c shall be taken –d times. We give then the following definition. "The operation of Multiplication is such that the product of the factors a-b and c-d will be equivalent to ac-ad-bc+bd, whatever may be the values of a, b, c, d." 7. 3a3+4a2-5a by -2a3. 8. -a3-a3-a by -α-1. 9. 3x3y-5xy2+4y3 by −2x-3y. III. INVOLUTION. 59. To this part of Algebra belongs the process called Involution. This is the operation of multiplying a quantity by itself any number of times. The power to which the quantity is raised is expressed by the number of times the quantity has been employed as a factor in the operation. Thus, as has been already stated in Art. 45, a2 is called the second power of a, a is called the third power of a. 60. When we have to raise negative quantities to certain powers we symbolize the operation by putting the quantity in a bracket with the letter denoting the index (Art. 45) placed over the bracket on the right hand. Thus (-a)3 denotes the third power of -a, (− 2x)a denotes the fourth power of −2x. 61. The signs of all even powers of a negative quantity will be positive, and the signs of the odd powers will be negative. (− a)3 = ( − a). (—a). (− a) = a2. ( − a) = — a3. 62. To raise a simple quantity to any power we multiply the index of the quantity by the number denoting the power to which it is to be raised, and prefix the proper sign. Thus the square of a3 is a®, the cube of as is a9, the cube of x2yz3 is — x¤ý3zo. 63. We form the second, third and fourth powers of a+b in the following manner: a+b a+b a2 + ab +ab +b2 (a+b)3=a2+2ab+b2 a + b a3+2a2b+ab + a2b+2ab2+b3 (a+b)3 = a3 +3a2b+3al2+b3 a + b a*+3a3b+3a2b2 + ab3 + a3b+3a2ba +3ab3 +ba (a+b) = a*+4a3b+6a2b2+4ab3+b* Here observe the following laws : 1. The indices of a decrease by unity in each term. 64. We form the second, third and fourth powers of a-b in the following manner: |