Transformation of Equations and Reciprocal Equations - Question Bank
1. The sum of the roots of a reciprocal equation of the first type of degree n is always:
2. If α, β, γ are the roots of x³ - 2x² + 3x - 4 = 0, find the equation whose roots are α-1, β-1, γ-1.
3. Which type of equation is x⁴ - 5x³ + 6x² - 5x + 1 = 0?
4. What is the sum of the roots of the equation formed by replacing x with (x+2) in x³ - 4x² + 5x - 1 = 0?
5. If α, β are the roots of x² - 3x + 2 = 0, find the equation whose roots are α² and β².
6. For a reciprocal equation of the second type of even degree, like ax⁴ + bx³ + cx² - bx + a = 0, what is a guaranteed root?
7. If α, β, γ are roots of x³ + 6x² + 11x + 6 = 0, find the equation whose roots are α+1, β+1, γ+1.
8. The reciprocal equation x⁵ + 1 = 0 has roots:
9. Transform the equation x³ - 3x² + 2x - 1 = 0 into an equation whose roots are 3 times the roots of the original equation.
10. If α, β are roots of x² + 5x + 6 = 0, find the equation whose roots are α-2 and β-2.
11. The equation x³ - 2x² - 2x + 1 = 0 is a reciprocal equation of:
12. If α, β, γ are roots of x³ + px² + qx + r = 0, the equation whose roots are α+β, β+γ, γ+α is:
13. Find the equation whose roots are the negatives of the roots of x³ - 2x² + 3x - 4 = 0.
14. For the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the first type, which condition MUST hold?
15. If α, β, γ are roots of x³ + 2x² + 3x + 4 = 0, find the equation whose roots are α+1, β+1, γ+1.
16. The equation 3x⁴ - 5x³ + 7x² - 5x + 3 = 0 is a reciprocal equation of:
17. What is the sum of the roots of the equation obtained by replacing x with (x-1) in x³ - 5x² + 6x - 1 = 0?
18. If α, β, γ are the roots of x³ + 2x + 1 = 0, find the equation whose roots are α², β², γ².
19. The equation x⁵ - 3x⁴ + 2x³ - 2x² + 3x - 1 = 0 is a reciprocal equation of:
20. If α, β are the roots of x² - 6x + 8 = 0, find the equation whose roots are α+3 and β+3.
21. Consider the reciprocal equation x⁴ + 3x³ + 4x² + 3x + 1 = 0. After dividing by x² and substituting y = x + 1/x, the equation in y is:
22. If α, β, γ are the roots of x³ + 2x² - x + 3 = 0, find the equation whose roots are 1/α, 1/β, 1/γ.
23. The equation 2x⁵ - 3x⁴ + 4x³ - 4x² + 3x - 2 = 0 is a reciprocal equation of:
24. What is the transformed equation if we replace x by (x+1) in the equation x³ - 2x² + 3x - 4 = 0?
25. If α, β are the roots of x² - 4x + 3 = 0, find the equation whose roots are α³ and β³.
26. Transform the equation x³ + 6x² + 11x + 6 = 0 into an equation whose roots are the reciprocals of its roots.
27. A reciprocal equation of the second type of odd degree always has a root:
28. If the roots of ax² + bx + c = 0 are α and β, the equation with roots α+1/α and β+1/β is:
29. If α, β, γ are roots of x³ - 6x² + 11x - 6 = 0, find the equation whose roots are α-1, β-1, γ-1.
30. The equation x³ + 2x² + 2x + 1 = 0 is a reciprocal equation of:
31. If the roots of x² + ax + b = 0 are α and β, find the equation whose roots are α+k and β+k.
32. Solve the reciprocal equation x⁴ - 5x³ + 6x² - 5x + 1 = 0.
33. What is the relationship between the coefficients of a reciprocal equation of the first type of odd degree?
34. If α, β, γ are the roots of x³ + px² + qx + r = 0, the equation whose roots are αβ, βγ, γα is:
35. Find the equation whose roots are the reciprocals of the roots of 2x³ - 3x² + 4x - 5 = 0.
36. If y = x + 1/x, then x² + 1/x² can be expressed in terms of y as:
37. The equation x⁵ + 2x⁴ + 3x³ + 3x² + 2x + 1 = 0 is:
38. If α, β, γ are the roots of x³ - 3x² + 2x - 1 = 0, find the equation whose roots are α+2, β+2, γ+2.
39. Given the equation x² - 5x + 6 = 0 with roots α and β, find the equation whose roots are 1/α and 1/β.
40. What is the condition for the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the second type?
41. If the roots of x³ + 2x² + 3x + 4 = 0 are α, β, γ, what is the sum of the roots of the equation whose roots are 2α, 2β, 2γ?
42. To solve a reciprocal equation of the form ax⁴ + bx³ + cx² + bx + a = 0, we typically divide by x² and substitute:
43. Consider the equation x⁴ + 2x³ + 3x² + 2x + 1 = 0. This is an example of:
44. If α, β are the roots of x² + bx + c = 0, then the equation with roots α² and β² is:
45. Find the transformed equation whose roots are the squares of the roots of the equation x³ - 6x² + 11x - 6 = 0.
46. What is the condition for the equation ax³ + bx² + cx + d = 0 to be a reciprocal equation of the first type?
47. If α, β, γ are the roots of the cubic equation ax³ + bx² + cx + d = 0, what is the sum of the roots of the equation formed by replacing x with (1/x)?
48. Given the equation x² + 5x + 6 = 0 with roots α and β, find the equation whose roots are α+1 and β+1.
49. If the roots of the equation x³ + px² + qx + r = 0 are α, β, γ, what are the roots of the equation x³ - px² + qx - r = 0?