46. We shall now give a few examples of greater difficulty. Example 1. Find the product of 3x2 - 2x-5 and 2x - 5. 47. If the expressions are not arranged according to powers, ascending or descending, of some common letter, a rearrangement will be found convenient. Example. Find the product of 2a2+4b2-3ab and 3ab - 5a2 +462. 15. x2-xy+y2, x2+xy+y2. 16. 17. a2-b2-3c2, -a2-b2-3c2. 18. 19. a3-6a+5, a3+6a-5. a2-2ax+2x2, a2+2ax+2x2. x3-3x2- x, x2 − 3x+1. 20. 2y+-4y2+1, 2y4 - 4y2 - 1. 21. 5m2+3-4m, 5-4m+3m2. 22. Sa3 - 2a2 - 3a, 3a2+1-5a. 23. 2x+2x3-3x2, 3x+2+2x2. 24. a3+b3-a2b2, a2b2 − a3+b3. 25. a3+x+3ax2+3a2x, a3 +3αx2-x3-3a2x. 28. a1+1+6a2 - 4a3 -- 4a, a3 − 1 + 3a - 3a2. 29. a2+b2+c2+ab+ac-bc, a-b-c. 30. x1+6x2y2+ ya − 4x3y − 4xy3, − x1 — ya — 6x2y2 – 4xy3 – 4x3y. 48. Although the result of multiplying together two binomial factors, such as x+8 and x--7, can always be obtained by the methods already explained, it is of the utmost importance that the student should soon learn to write down the product rapidly by inspection. This is done by observing in what way the coefficients of the terms in the product arise, and noticing that they result from the combination of the numerical coefficients in the two binomials which are multiplied together; thus (x+8)(x+7)=x2+8x+7x+56 = x2+15x+56. (x −8)(x −7) = x2 – 8x − 7x+56 (x+8)(x-7)=x2+8x-7x-56 = x2+x-56. (x−8)(x+7)= x2 - 8x+7x-56 =x2-x-56. In each of these results we notice that: 1. The product consists of three terms. 2. The first term is the product of the first terms of the two binomial expressions. 3. The third term is the product of the second terms of the two binomial expressions. 4. The middle term has for its coefficient the sum of the numerical quantities (taken with their proper signs) in the second terms of the two binomial expressions. The intermediate step in the work may be omitted, and the products written down at once, as in the following examples : (+2(+3)=+5 +6. (x − 3)(x+4)= x2+x−12. By an easy extension of these principles the product of any two binomials. we may write down Thus (2r+3y)(x-y)=2x+3.ry-2xy-3y2 (3-4)(2x+y)=6.x2-8xy+3xy-4y2 (x+4)(x-4)=x+4x-4x-16 (2x+5y)(2x-5y)=4x+10xy-10xy-25y2 28. (3x+2a)(3x-2a). 29. (6x+a) (6x-2a). 30. (7x+3y)(7x − y). CHAPTER VI. DIVISION. 49. THE object of division is to find out the quantity, called the quotient, by which the divisor must be multiplied so as to produce the dividend. or Division is thus the inverse of multiplication. The above statement may be briefly written quotient × divisor=dividend, It is sometimes better to express this last result as a frac tion; thus dividend =quotient. Example 1. divisor Since the product of 4 and x is 4x, it follows that when 4x is divided by x the quotient is 4, In each of these cases it should be noticed that the index of any letter in the quotient is the difference of the indices of that letter in the dividend and divisor. 50. It is easy to prove that the rule of signs holds for division. Hence in division as well as multiplication like signs produce +, unlike signs produce Rule. To divide one simple expression by another : The index of each letter in the quotient is obtained by subtracting the index of that letter in the divisor from that in the dividend. To the result so obtained prefix with its proper sign the quotient of the coefficient of the dividend by that of the divisor. Example 1. Divide 84a5x3 by - 12a1x. The quotient = ( − 7) × a5-4x3-1 =-7αx2 Or at once mentally, 84a5x3(-12a1x) = − 7ax2. Example 2. - 45a®b2x1÷( ~ 9a3bx2) = 5a3bx2. Note. If we apply the rule to divide any power of a letter by the same power of the letter we are led to a curious conclusion. Thus, by the rule but also a3÷a3 = a3-3 = ao ; This result will appear somewhat strange to the beginner, but its full significance is explained in the Theory of Indices. [See Elementary Algebra, Chap. xxxI.] Rule. To divide a compound expression by a single factor, divide each term separately by that factor, and take the algebraic sum of the partial quotients so obtained. This follows at once from Art. 38. Examples. (9x - 12y+32)÷(−3) : = -3x+4y - %. (36a3b2-24a2b5 - 20a1b2)÷4a2b = 9ab6b4 -- 5a2b. |