12. Divide a1+3y a3+ a2+5y by a2-x+2y. 13. Divide x - xm-5n+4 + x2m by x2-m-2n. 14. Divide yp - y3-4m + y+p+1 by y2p-m+1. 15. Divide 2 x3n - 6 x2nyn + 6 xny2n 2 y3n by xn yn. 16. Divide x 2 ax2 + a2x - abx - bx + a2b + ab2 21. Arrange according to descending powers of x the. following expression, and enclose the coefficient of each power in a parenthesis with a minus sign before each parenthesis except the first: 22. Divide 1.2 aʻx – 5.494 a3x2 + 4.8 a2x3 + 0.9 ax2 - x5 by 0.6 ax 2 x2. 23. Multiply a2-ab+b2 bya + b. 24. Multiply a2 + ab + b2 bya-b. 25. Divide a3+ + ab2 + b2 by a + b. 26. Subtract + x2 + xy + y2 from x2 - + xy + y2. 27. Subtract x2 + xy - + y2 from 2x2 xy + y2. 28. If a = 8, b = 6, c = - 4, find the value of √a2+2bc+b2 + ac+c2 + ab. CHAPTER VI. SPECIAL RULES. Multiplication. 107. Square of the Sum of Two Numbers. (a + b)2= (a + b) (a + b) = a (a + b) + b (a + b) RULE 1. The square of the sum of two numbers is the sum of their squares plus twice their product. 108. Square of the Difference of Two Numbers. (a - b)2 = (a - b) (a - b) = a (a - b) - b (a - b) RULE 2. The square of the difference of two numbers ic the sum of their squares minus twice their product. 109. Product of the Sum and Difference of Two Numbers. (a + b) (a - b) = a (a - b) + b (a - b) RULE 3. The product of the sum and difference of two numbers is the difference of their squares. 110. The following rule for raising a monomial to any required power will be useful in solving examples in multiplication: Raise the numerical coefficient to the required power, and multiply the exponent of each letter by the exponent of the required power. Thus the square of 7 a266 is 49 a4612. 111. If we are required to multiply a + b + c by a+b-c, we may abridge the ordinary process as follows: (a + b + c) (a + b - c) = {(a + b)+c}{(a + b) - c} By Rule 3, By Rule 1, = (a + b)2 - c2 = a2 + 2ab + b2 - c2. If we are required to multiply a + b -c by a - b + c, we may put the expressions in the following forms, and perform the operation : (a + b - c) (a - b + c) = {a + (b - c)}{a - (b - c)} 11.1 + x + y and 1 + x 12. а2 - 2 ax + 4x2 and a2 + 2 ax + 4 x2. a + 1. z and 3x 2y + z. y. 112. Square of any Polynomial. If we put x for a, and y + z for b, in the identity we have (a + b)2 = a2 + 2ab + b2, {x + (y + z) }2 = x2 + 2 x (y + z) + (y + z)2, or (x + y + z) 2 = x2 + 2xy + 2 xz + y2 + 2 yz + z2 = x2 + y2 + z2 + 2xy + 2 xz + 2 yz. The complete product consists of the sum of the squares of the terms of the given expression and twice the product of each term into all the terms that follow it. Again, if we put a - b for a, and c d for b, in the same identity, we have = (a - b)2 + 2 (a - b) (c-d) + (c - d)2 = (a2-2ab+b2) + 2a (c-d) - 2b (c-d) + (c2 - 2cd+d2) = a2-2ab+b2+2ac-2 ad - 2bc+2bd+c2 - 2cd+d2 = a2 + b2+c2+d2-2ab+2ac-2ad - 2bc + 2bd – 2cd. Here the same law holds as before, the sign of each double product being + or -, according as the factors composing it have like or unlike signs. The same is true for any polynomial. Hence we have the following rule: RULE 4. The square of a polynomial is the sum of the squares of the several terms and twice the product obtained by multiplying each term into all the terms that follow it. |