x+2y x Miscellaneous, 1. Prove the rule for dividing one algebraical fraction by another. x+y 4. Give the rules for multiplying and dividing algebraical fractions, and express in their simplest forms the product of ax a2-x2 (a-x)2 and ab and the quotient of 2ax-x2 2α-x SQUARE ROOT. 1. Find the square root of 4x2-24xy+362. 2. 4x4 4x3-3x2+2x+1. 8. 9a+12a3+22a2+12a +9. 17. Divide a2b-bx2+a2x-x3 by (a-x) (b+x). 18. Divide 1+ 2x by 1+ 3x to six terms. 19. Divide a1-a3 + 11a3 — a by a2 – ža. 20. Divide I -x by 1−x+x2 to four terms. 21. Divide 1-9x3- 8x9 by 1+2x+x2. 22. Divide a1+4a2x2+16x1 by a2+2ax +4x2. What does the quotient become when x=2? 2 23. Divide 1-a3+8b3+6ab by 1+2b-a. If a = 2, b=1, what is your result? 24. Divide 4a*b*+1 by 2a2b2 — 2ab+1, and prove the results when a = 2 and b = −3. 25. Divide x-a6 by the continued product of (x-4), (x+a), and (x2+ax+a2). 26. Divide x4-9x2-6xy-y2 by x2+3x+y, and ax3-a3x +xm-a2xm-2 by x-a. MISCELLANEOUS ON ADDITION, SUBTRACTION, 1. Add together a2-ab+b2, and 2a2+ 3ab-462, and subtract 3a2-4ab+562 from the result. 2. Express in the simplest form, 7x2 − 912 + 20xy + 103a - 11x2-6xy+8x2 — 73a — 4xy. 3. Add together 3x2y2-1014, -x2y2 + 514, 8x2y2 −6y+, and 4x2,y2+214; and from their sum take 10x22-4x4 — 51a. 4. Find the value of x3- 2x2y-3xy2-413 when y=-1, x=3, and subtract 2a-3(2b-3c) from 4a-3(b-2c). 5. Add together a(x+y)+b(y+s), and a(x-y)-by-s); and also subtract the latter from the former. 6. Simplify +*+Zx3⁄4y − Zx2y2+ $y* − {x2y2 + }x+− {x}, and find its value when x=7 and y = 1. 7. Subtract (b-a) (c–d) from (a−b) (c–d), and give the value of the result when a = 2b, d= 2c. 8. Add together 3x+6y-72, 2x-7y+5%, and x+5+5 and subtract 2x-3(x-(y-x)} from 2y-3-(x − y)}. 11. Show that a quantity may be transferred from one side of an equation to the other by changing its sign without destroying the equality expressed by it. 12. Solve the equations— (1) 9x-8 (2) 7 x+2 2 + = 0. 4 9 14. Solve the equations— (1) 3(x-5)-5(x-4)=21x-41. (2) 5x — 1 _ 7x-2=63_1 2 ΙΟ 15. Solve the equations 5 2 16. Find the value of x in the equation 5x-7 3x-4_9-2x = 27. Divide the product of x3-12x-16 and x3-12x+16 by x8x2+16. 28. Divide + 3ax3 + a3x + 3a1 by x2+4ax + 3a2, and multiply the quotient by x2+ax+a2. m 29. Divide am+" — xTMy" +x*y* —"+" by x*-*, and multiply the quotient by x-y. MISCELLANEOUS (on the above generally). x √ x 1. Find the value of 1+ when x and y = }. y I-y = 2. Find the value of x2+(x2-42x+89) when x=2; also 3. (a) By how much is x greater than 50? (b) If I travel x miles at the rate of y miles an hour, what is the number of hours? (c) Resolve x2+6ax-91a2 into factors. 4. Find the value of (2+3x+4x2)2− (2 −3x+4x2)9. 5. What is the meaning of a term, a coefficient, an index, a negative quantity, and a factor? Give an example of each, and resolve 12x2+xy-632 into factors. 6. What is a factor? Resolve as+63 +3(a+b)ba into factors. 7. Prove that (x2+xy+y2)2= (x2−xy+y2)2+ 4xy(x2+ƒ3) ; and show that the equation holds good if x=5 and y = −5. 8. Show that x(x+1) (x+2) (x+3)+1 = (x2+3x+1)2. 9. Show that if the sum of any two quantities divide the difference of their squares, the quotient is equal to the difierence of the two quantities. Divide (a2x2+b22) — (a2b2+x2y2) by ax+by+ab+xy. 10. (1) What is the sum of a+a+a+written times? (2) Show that the square of the sum of any two consecutive integers is greater by one than four times their product. 11. Find the value of x-(a-b)x3 + (a−b)b3x − b1, when a=-x=},b=0. |