Equation with linearly transformed roots

AQA Further Paper 2 2021 June Q5

5 marks Standard +0.3

5 The equation $$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$ has roots $\alpha , \beta$ and $\gamma$ Find a cubic equation with roots $$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$

OCR FP1 AS 2021 June Q1

3 marks Standard +0.3

1 In this question you must show detailed reasoning.
The cubic equation $2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$. By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.

OCR MEI FP1 2007 June Q5

6 marks Standard +0.3

The roots of the cubic equation $x^3 + 3x^2 - 7x + 1 = 0$ are $\alpha$, $\beta$ and $\gamma$. Find the cubic equation whose roots are $3\alpha$, $3\beta$ and $3\gamma$, expressing your answer in a form with integer coefficients. [6]

AQA Further AS Paper 1 2019 June Q14

7 marks Standard +0.8

The graph of $y = x^3 - 3x$ is shown below. \includegraphics{figure_14} The two stationary points have $x$-coordinates of $-1$ and $1$ The cubic equation $$x^3 - 3x + p = 0$$ where $p$ is a real constant, has the roots $\alpha$, $\beta$ and $\gamma$. The roots $\alpha$ and $\beta$ are not real.

Explain why $\alpha + \beta = -\gamma$ [1 mark]
Find the set of possible values for the real constant $p$. [2 marks]
$f(x) = 0$ is a cubic equation with roots $\alpha + 1$, $\beta + 1$ and $\gamma + 1$
1. Show that the constant term of $f(x)$ is $p + 2$ [3 marks]
2. Write down the $x$-coordinates of the stationary points of $y = f(x)$ [1 mark]

AQA Further Paper 2 2024 June Q6

3 marks Moderate -0.3

The cubic equation $$x^3 + 5x^2 - 4x + 2 = 0$$ has roots $\alpha$, $\beta$ and $\gamma$ Find a cubic equation, with integer coefficients, whose roots are $3\alpha$, $3\beta$ and $3\gamma$ [3 marks]

Edexcel CP1 2021 June Q3

6 marks Standard +0.8

The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where $a$ and $b$ are constants, has roots $\alpha$, $\beta$ and $\gamma$ The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where $c$ is a constant, has roots $(4\alpha - 1)$, $(4\beta - 1)$ and $(4\gamma - 1)$ Without solving either cubic equation, determine the value of $a$, the value of $b$ and the value of $c$. [6]

OCR Further Pure Core AS 2020 November Q5

7 marks Challenging +1.8

In this question you must show detailed reasoning. The cubic equation $5x^3 + 3x^2 - 4x + 7 = 0$ has roots $\alpha$, $\beta$ and $\gamma$. Find a cubic equation with integer coefficients whose roots are $\alpha + \beta$, $\beta + \gamma$ and $\gamma + \alpha$. [7]

SPS SPS FM Pure 2023 February Q8

6 marks Challenging +1.8

The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where $a$ and $b$ are constants, has roots $\alpha$, $\beta$ and $\gamma$ The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where $c$ is a constant, has roots $(4\alpha - 1)$, $(4\beta - 1)$ and $(4\gamma - 1)$ Without solving either cubic equation, determine the value of $a$, the value of $b$ and the value of $c$. [6]

SPS SPS FM Pure 2023 November Q3

8 marks Standard +0.8

In this question you must show detailed reasoning. The roots of the equation $2x^3 - 5x + 7 = 0$ are $\alpha$, $\beta$ and $\gamma$.

Find $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$. [4]
Find an equation with integer coefficients whose roots are $2\alpha - 1$, $2\beta - 1$ and $2\gamma - 1$. [4]

SPS SPS FM Pure 2025 February Q4

5 marks Standard +0.3

The cubic equation $$2x^3 + 6x^2 - 3x + 12 = 0$$ has roots $\alpha$, $\beta$ and $\gamma$. Without solving the equation, find the cubic equation whose roots are $(\alpha + 3)$, $(\beta + 3)$ and $(\gamma + 3)$, giving your answer in the form $pw^3 + qw^2 + rw + s = 0$, where $p$, $q$, $r$ and $s$ are integers to be found. [5]