9231/1 Roots of Polynomial Equations
CAIE/9231/01/MJ02/Q5
The roots of the equation x³ - 3x² + 1 = 0 are denoted by α, β, γ. Show that the equation whose roots are α/(α-2), β/(β-2), γ/(γ-2) is 3y³ - 9y² - 3y + 1 = 0.
Find the value of:
- (i) (α - 2)(β - 2)(γ - 2) [3 marks]
- (ii) α(β - 2)(γ - 2) + β(γ - 2)(α - 2) + γ(α - 2)(β - 2) [2 marks]
Answers: (i) 3; (ii) 9
CAIE/9231/01/ON02/Q2
The equation x⁴ + Ax³ + Bx² + 4x - 2 = 0, where A and B are constants, has roots α, β, γ, δ. Find a polynomial equation whose roots are 1/α, 1/β, 1/γ, 1/δ. [2 marks]
Given that α² + β² + γ² + δ² = 1/α² + 1/β² + 1/γ² + 1/δ², find the value of A. [3 marks]
Context/Solution: 2u⁴ - 4u³ - Au² - u - 1 = 0; A = -1
CAIE/9231/01/MJ03/Q2
The equation 8x³ + 12x² + 4x - 1 = 0 has roots α, β, γ. Show that the equation with roots 2α + 1, 2β + 1, 2γ + 1 is y³ - y - 1 = 0. [3 marks]
The sum (2α + 1)ⁿ + (2β + 1)ⁿ + (2γ + 1)ⁿ is denoted by Sₙ. Find the values of S₃ and S₂. [5 marks]
Answers: S₃ = 3, S₂ = 1
CAIE/9231/01/ON03/Q6
Find the sum of the squares of the roots of the equation x³ + x + 12 = 0, and deduce that only one of the roots is real. [4 marks]
The real root of the equation is denoted by α. Prove that -3 < α < -2, and hence prove that the modulus of each of the other roots lies between 2 and √6. [5 marks]
Answer: Σα² = -2
CAIE/9231/01/MJ04/Q11O
The roots of the equation x³ - x - 1 = 0 are α, β, γ, and Sₙ = αⁿ + βⁿ + γⁿ.
(i) Use the relation y = x² to show that α², β², γ² are the roots of the equation y³ - 2y² + y - 1 = 0. [3 marks]
(ii) Hence, or otherwise, find the value of S₄. [2 marks]
(iii) Find the values of S₈, S₁₂ and S₁₆. [9 marks]
Answers: (ii) S₄ = 2; (iii) S₈ = 10, S₁₂ = 29, S₁₆ = 90
CAIE/9231/01/ON04/Q3
Given that α + β + γ = 0, α² + β² + γ² = 14, α³ + β³ + γ³ = -18, find a cubic equation whose roots are α, β, γ. [4 marks]
Hence find possible values for α, β, γ. [2 marks]
Answer: x³ - 7x + 6 = 0; 1, 2, -3
CAIE/9231/01/MJ05/Q4
Show that the sum of the cubes of the roots of the equation x³ + λx + 1 = 0 is -3. [3 marks]
Show also that there is no real value of λ for which the sum of the fourth powers of the roots is negative. [3 marks]
CAIE/9231/01/ON05/Q5
In the equation x³ + ax² + bx + c = 0, the coefficients a, b and c are real. It is given that all the roots are real and greater than 1.
(i) Prove that a < -3. [1 mark]
(ii) By considering the sum of the squares of the roots, prove that a² > 2b + 3. [2 marks]
(iii) By considering the sum of the cubes of the roots, prove that a³ < -9b - 3c - 3. [4 marks]
CAIE/9231/01/MJ06/Q11E
Obtain the sum of the squares of the roots of the equation x⁴ + 3x³ + 5x² + 12x + 4 = 0. [2 marks]
Deduce that this equation does not have more than 2 real roots. [3 marks]
Show that, in fact, the equation has exactly 2 real roots in the interval -3 < x < 0. [5 marks]
Denoting these roots by α and β, and the other 2 roots by γ and δ, show that |γ| = |δ| = 2 / √(αβ). [4 marks]
CAIE/9231/01/ON06/Q6
The roots of the equation x³ + x + 1 = 0 are α, β, γ. Show that the equation whose roots are (4α + 1)/(α + 1), (4β + 1)/(β + 1), (4γ + 1)/(γ + 1) is of the form y³ + py + q = 0, where the numbers p and q are to be determined. [5 marks]
Hence find the value of (4α + 1)ⁿ/(α + 1)ⁿ + (4β + 1)ⁿ/(β + 1)ⁿ + (4γ + 1)ⁿ/(γ + 1)ⁿ, for n = 2 and for n = 3. [4 marks]
Answers: p = -21, q = 47; 42, -141
CAIE/9231/01/MJ07/Q7
The equation x³ + 3x - 1 = 0 has roots α, β, γ. Use the substitution y = x³ to show that the equation whose roots are α³, β³, γ³ is y³ - 3y² + 30y - 1 = 0. [2 marks]
Find the value of α⁹ + β⁹ + γ⁹. [5 marks]
Answer: -240
CAIE/9231/01/ON07/Q4
The roots of the equation x³ - 8x² + 5 = 0 are α, β, γ. Show that α² = 5 / (βγ). [4 marks]
It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive. [3 marks]
CAIE/9231/01/MJ08/Q5
The equation x³ + x - 1 = 0 has roots α, β, γ. Show that the equation with roots α³, β³, γ³ is y³ - 3y² + 4y - 1 = 0. [4 marks]
Hence find the value of α⁶ + β⁶ + γ⁶. [3 marks]
Answer: 1
CAIE/9231/01/ON08/Q12O
The roots of the equation x⁴ - 5x³ + 2x - 1 = 0 are α, β, γ, δ. Let Sₙ = αⁿ + βⁿ + γⁿ + δⁿ.
(i) Show that S<0xE2><0x82><0x99>₊₄ - 5S<0xE2><0x82><0x99>₊₃ + 2S<0xE2><0x82><0x99>₊₂ - S<0xE2><0x82><0x99>₊₁ - Sₙ = 0. [2 marks]
(ii) Find the values of S₂ and S₄. [3 marks]
(iii) Find the value of S₃ and hence find the value of S₆. [6 marks]
(iv) Hence find the value of α²(β⁴ + γ⁴ + δ⁴) + β²(γ⁴ + δ⁴ + α⁴) + γ²(δ⁴ + α⁴ + β⁴) + δ²(α⁴ + β⁴ + γ⁴). [3 marks]
Answers: (ii) S₂ = 10, S₄ = 54; (iii) S₃ = -6, S₆ = 292; (iv) 248
CAIE/9231/01/MJ09/Q1
The equation x⁴ - x³ - 1 = 0 has roots α, β, γ, δ. By using the substitution y = x³, or by any other method, find the exact value of α⁶ + β⁶ + γ⁶ + δ⁶. [5 marks]
CAIE/9231/01/ON09/Q5
The equation x³ + 5x + 3 = 0 has roots α, β, γ. Use the substitution x = -3/y to find a cubic equation in y and show that the roots of this equation are βγ, γα, αβ. [4 marks]
Find the exact values of β²γ² + γ²α² + α²β² and β³γ³ + γ³α³ + α³β³. [5 marks]
Answers: y³ - 5y² - 9 = 0; 25, 152
CAIE/9231/11/12/MJ10/Q6
The equation x³ + x - 1 = 0 has roots α, β, γ. Use the relation x = √y to show that the equation y³ + 2y² + y - 1 = 0 has roots α², β², γ². [2 marks]
Let Sₙ = αⁿ + βⁿ + γⁿ.
(i) Write down the value of S₂ and show that S₄ = 2. [3 marks]
(ii) Find the values of S₆ and S₈. [4 marks]
Answers: (i) -2; (ii) S₆ = 1, S₈ = -6
CAIE/9231/13/MJ10/Q10
The equation x⁴ + x³ + cx² + 4x - 2 = 0, where c is a constant, has roots α, β, γ, δ.
(i) Use the substitution y = 1/x to find an equation which has roots 1/α, 1/β, 1/γ, 1/δ. [2 marks]
(ii) Find, in terms of c, the values of α² + β² + γ² + δ² and 1/α² + 1/β² + 1/γ² + 1/δ². [3 marks]
(iii) Hence find, in terms of c, (α - 1/α)² + (β - 1/β)² + (γ - 1/γ)² + (δ - 1/δ)². [2 marks]
(iv) Deduce that when c = -3 the roots of the given equation are not all real. [3 marks]
Answers: (i) 2y⁴ - 4y³ - cy² - y - 1 = 0; (ii) 1 - 2c, 4 + c; (iii) -c - 3
CAIE/9231/01/ON10/Q7
The roots of the equation x³ + 4x - 1 = 0 are α, β and γ. Use the substitution y = 1/(1 + x) to show that the equation 6y³ - 7y² + 3y - 1 = 0 has roots 1/(α + 1), 1/(β + 1) and 1/(γ + 1). [2 marks]
For the cases n = 1 and n = 2, find the value of 1/(α + 1)ⁿ + 1/(β + 1)ⁿ + 1/(γ + 1)ⁿ. [2 marks]
Deduce the value of 1/(α + 1)³ + 1/(β + 1)³ + 1/(γ + 1)³. [2 marks]
Hence show that (β + 1)(γ + 1) / (α + 1)² + (γ + 1)(α + 1) / (β + 1)² + (α + 1)(β + 1) / (γ + 1)² = 73/36. [3 marks]
Answers: 7/6, 13/36; 73/216
CAIE/9231/11/12/MJ11/Q2
The roots of the equation x³ + px² + qx + r = 0 are β/k, β, kβ, where p, q, r, k and β are non-zero real constants. Show that β = -q/p. [4 marks]
Deduce that rp³ = q³. [2 marks]
CAIE/9231/13/MJ11/Q3
Find a cubic equation with roots α, β and γ, given that α + β + γ = -6, α² + β² + γ² = 38, αβγ = 30. [3 marks]
Hence find the numerical values of the roots. [3 marks]
Answer: t³ + 6t² - t - 30 = 0; 2, -3, -5
CAIE/9231/11/12/ON11/Q1
The equation x³ + px + q = 0 has a repeated root. Prove that 4p³ + 27q² = 0. [5 marks]
CAIE/9231/13/ON11/Q3
The equation x³ + 5x² - 3x - 15 = 0 has roots α, β, γ. Find the value of α² + β² + γ². [3 marks]
Hence show that the matrix [[1, α, β], [β, 1, γ], [γ, β, 1]] is singular. [4 marks]
Answer: 31
CAIE/9231/11/12/MJ12/Q1
The roots of the cubic equation x³ - 7x² + 2x - 3 = 0 are α, β, γ. Find the values of:
(i) α² + β² + γ², [2 marks]
(ii) α³ + β³ + γ³. [3 marks]
Answers: (i) 45; (ii) 310