Transformation of Equations and Reciprocal Equations - One Line Questions
1.
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? —
-8
2.
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)? —
-b/d
3.
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? —
-α, -β, -γ
4.
The sum of the roots of a reciprocal equation of the first type of degree n is always: —
0
5.
A reciprocal equation of the second type of odd degree always has a root: —
-1
6.
For a reciprocal equation of the second type of even degree, like ax⁴ + bx³ + cx² - bx + a = 0, what is a guaranteed root? —
-1
7.
What is the sum of the roots of the equation formed by replacing x with (x+2) in x³ - 4x² + 5x - 1 = 0? —
10
8.
What is the sum of the roots of the equation obtained by replacing x with (x-1) in x³ - 5x² + 6x - 1 = 0? —
4
9.
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. —
3x³ - x² + 2x + 1 = 0
10.
Find the equation whose roots are the reciprocals of the roots of 2x³ - 3x² + 4x - 5 = 0. —
5x³ - 4x² + 3x - 2 = 0
11.
Given the equation x² - 5x + 6 = 0 with roots α and β, find the equation whose roots are 1/α and 1/β. —
6x² - 5x + 1 = 0
12.
Transform the equation x³ + 6x² + 11x + 6 = 0 into an equation whose roots are the reciprocals of its roots. —
6x³ + 11x² + 6x + 1 = 0
13.
What is the condition for the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the second type? —
a = -e and b = d
14.
What is the condition for the equation ax³ + bx² + cx + d = 0 to be a reciprocal equation of the first type? —
a = d and b = c
15.
For the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the first type, which condition MUST hold? —
a = e and b = d
16.
Consider the equation x⁴ + 2x³ + 3x² + 2x + 1 = 0. This is an example of: —
A reciprocal equation of the first type
17.
The equation x⁵ + 2x⁴ + 3x³ + 3x² + 2x + 1 = 0 is: —
A reciprocal equation of the second type (odd degree)
18.
If the roots of ax² + bx + c = 0 are α and β, the equation with roots α+1/α and β+1/β is: —
a²x² - (ab-bc)x + (b²-ac) = 0
19.
What is the relationship between the coefficients of a reciprocal equation of the first type of odd degree? —
Coefficients equidistant from the beginning and end are equal
20.
The equation 3x⁴ - 5x³ + 7x² - 5x + 3 = 0 is a reciprocal equation of: —
First type, even degree
21.
The equation x³ + 2x² + 2x + 1 = 0 is a reciprocal equation of: —
First type, odd degree
22.
The equation 2x⁵ - 3x⁴ + 4x³ - 4x² + 3x - 2 = 0 is a reciprocal equation of: —
First type, odd degree
23.
The equation x⁵ - 3x⁴ + 2x³ - 2x² + 3x - 1 = 0 is a reciprocal equation of: —
Second type, odd degree
24.
Which type of equation is x⁴ - 5x³ + 6x² - 5x + 1 = 0? —
Reciprocal equation of the first type, even degree
25.
Solve the reciprocal equation x⁴ - 5x³ + 6x² - 5x + 1 = 0. —
Roots are 1, 1, (5 ± √21)/2
26.
The equation x³ - 2x² - 2x + 1 = 0 is a reciprocal equation of: —
Second type, odd degree
27.
The reciprocal equation x⁵ + 1 = 0 has roots: —
The roots of x+1=0 and the roots of x⁴+x³+x²+x+1=0
28.
If α, β are the roots of x² + bx + c = 0, then the equation with roots α² and β² is: —
x² - (b² - 2c)x + c² = 0
29.
If α, β are the roots of x² - 6x + 8 = 0, find the equation whose roots are α+3 and β+3. —
x² - 12x + 24 = 0
30.
If α, β are the roots of x² - 4x + 3 = 0, find the equation whose roots are α³ and β³. —
x² - 37x + 27 = 0
31.
If α, β are the roots of x² - 3x + 2 = 0, find the equation whose roots are α² and β². —
x² - 5x + 4 = 0
32.
If the roots of x² + ax + b = 0 are α and β, find the equation whose roots are α+k and β+k. —
x² + (a-2k)x + (b-ak+k²) = 0
33.
Given the equation x² + 5x + 6 = 0 with roots α and β, find the equation whose roots are α+1 and β+1. —
x² + 3x + 2 = 0
34.
If α, β are roots of x² + 5x + 6 = 0, find the equation whose roots are α-2 and β-2. —
x² + x + 2 = 0
35.
If α, β, γ are roots of x³ - 6x² + 11x - 6 = 0, find the equation whose roots are α-1, β-1, γ-1. —
x³ - 3x² + 2x = 0
36.
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. —
x³ - 5x² + 7x - 10 = 0
37.
Transform the equation x³ - 3x² + 2x - 1 = 0 into an equation whose roots are 3 times the roots of the original equation. —
x³ - 9x² + 18x - 27 = 0
38.
If α, β, γ are the roots of x³ + px² + qx + r = 0, the equation whose roots are αβ, βγ, γα is: —
x³ - qx² + (pr)x - r² = 0
39.
Find the equation whose roots are the negatives of the roots of x³ - 2x² + 3x - 4 = 0. —
x³ + 2x² + 3x + 4 = 0
40.
Find the transformed equation whose roots are the squares of the roots of the equation x³ - 6x² + 11x - 6 = 0. —
x³ - 30x² + 121x - 36 = 0
41.
If α, β, γ are roots of x³ + 6x² + 11x + 6 = 0, find the equation whose roots are α+1, β+1, γ+1. —
x³ + 3x² + 2x = 0
42.
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. —
x³ + 3x² + 4x + 1 = 0
43.
If α, β, γ are the roots of x³ + 2x + 1 = 0, find the equation whose roots are α², β², γ². —
x³ + 4x² - 4x - 1 = 0
44.
If α, β, γ are roots of x³ + 2x² + 3x + 4 = 0, find the equation whose roots are α+1, β+1, γ+1. —
x³ + 5x² + 8x + 5 = 0
45.
If α, β, γ are roots of x³ + px² + qx + r = 0, the equation whose roots are α+β, β+γ, γ+α is: —
x³ - px² + (q²-2pr)x - (r(p²-2q)) = 0
46.
What is the transformed equation if we replace x by (x+1) in the equation x³ - 2x² + 3x - 4 = 0? —
x³ + x² + 2x - 2 = 0
47.
To solve a reciprocal equation of the form ax⁴ + bx³ + cx² + bx + a = 0, we typically divide by x² and substitute: —
y = x + 1/x
48.
If y = x + 1/x, then x² + 1/x² can be expressed in terms of y as: —
y² - 2
49.
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: —
y² + 3y + 2 = 0