Factor theorem and finding roots - Roots of polynomials

CAIE P3 2010 November Q10

13 marks Standard +0.3

10 The polynomial $\mathrm { p } ( z )$ is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where $m$ is a constant. It is given that $( z + 2 )$ is a factor of $\mathrm { p } ( z )$.

Find the value of $m$.
Hence, showing all your working, find
(a) the three roots of the equation $\mathrm { p } ( z ) = 0$,
(b) the six roots of the equation $\mathrm { p } \left( z ^ { 2 } \right) = 0$.

CAIE P3 2023 June Q10

12 marks Standard +0.8

10 The polynomial $x ^ { 3 } + 5 x ^ { 2 } + 31 x + 75$ is denoted by $\mathrm { p } ( x )$.

Show that $( x + 3 )$ is a factor of $\mathrm { p } ( x )$.
Show that $z = - 1 + 2 \sqrt { 6 } \mathrm { i }$ is a root of $\mathrm { p } ( z ) = 0$.
Hence find the complex numbers $z$ which are roots of $\mathrm { p } \left( z ^ { 2 } \right) = 0$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Edexcel F1 2014 January Q4

9 marks Standard +0.3

4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$

Given that $x = - 4$ and $x = 3$ are roots of the equation $\mathrm { f } ( x ) = 0$, use algebra to solve $\mathrm { f } ( x ) = 0$ completely.
Show the four roots of $\mathrm { f } ( x ) = 0$ on a single Argand diagram.

Edexcel F1 2017 January Q1

5 marks Standard +0.8

$\mathrm { f } ( x ) = 2 ^ { x } - 10 \sin x - 2$, where $x$ is measured in radians

Show that $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, between 2 and 3
[0pt]
Use linear interpolation once on the interval [2,3] to find an approximation to $\alpha$. Give your answer to 3 decimal places.

Edexcel F1 2023 January Q3

10 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where $p$ is a constant.
Given that - 3 is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

determine the value of $p$
using algebra, solve $\mathrm { f } ( \mathrm { z } ) = 0$ completely, giving the roots in simplest form,
determine the modulus of the complex roots of $\mathrm { f } ( \mathrm { z } ) = 0$
show the roots of $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.

Edexcel F1 2024 June Q2

9 marks Standard +0.3

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that $z = 3$ is a root of the equation $f ( z ) = 0$

show that $p = - 87$
Use algebra to determine the other roots of $\mathrm { f } ( \mathrm { z } ) = 0$, giving your answers in simplest form. On an Argand diagram
- the root $z = 3$ is represented by the point $P$
- the other roots of $\mathrm { f } ( \mathrm { z } ) = 0$ are represented by the points $Q$ and $R$
- the number $z = - 9$ is represented by the point $S$
- Show on a single Argand diagram the positions of $P , Q , R$ and $S$
- Determine the perimeter of the quadrilateral $P Q S R$, giving your answer as a simplified surd.

Edexcel FP1 Q1

5 marks Moderate -0.8

1. $$\mathrm { f } ( x ) = 2 x ^ { 3 } - 8 x ^ { 2 } + 7 x - 3$$ Given that $x = 3$ is a solution of the equation $\mathrm { f } ( x ) = 0$, solve $\mathrm { f } ( x ) = 0$ completely.

Edexcel FP1 2010 June Q4

7 marks Moderate -0.8

4. $$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$ Given that $\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)$, where $a$ and $b$ are real constants,

find the value of $a$ and the value of $b$.
Find the three roots of $\mathrm { f } ( x ) = 0$.
Find the sum of the three roots of $\mathrm { f } ( x ) = 0$.

Edexcel FP1 2013 June Q3

7 marks Moderate -0.3

3. Given that $x = \frac { 1 } { 2 }$ is a root of the equation $$2 x ^ { 3 } - 9 x ^ { 2 } + k x - 13 = 0 , \quad k \in \mathbb { R }$$ find

the value of $k$,
the other 2 roots of the equation.

Edexcel FP1 2018 June Q1

6 marks Moderate -0.5

1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that $\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)$, where $a$ and $b$ are real constants,

find the value of $a$ and the value of $b$.
Hence use algebra to find the three roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

OCR MEI FP1 2008 January Q3

7 marks Moderate -0.8

3

Show that $z = 3$ is a root of the cubic equation $z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0$ and find the other roots.
Show the roots on an Argand diagram.

Edexcel AEA 2015 June Q2

9 marks Challenging +1.2

2．（a）Show that $( x + 1 )$ is a factor of $2 x ^ { 3 } + 3 x ^ { 2 } - 1$
（b）Solve the equation
（b）Solve the equation $$\sqrt { x ^ { 2 } + 2 x + 5 } = x + \sqrt { 2 x + 3 }$$

OCR MEI FP1 2012 January Q3

6 marks Moderate -0.3

3 Given that $z = 6$ is a root of the cubic equation $z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0$, find the value of $p$ and the other roots.

OCR MEI FP1 2009 June Q2

5 marks Moderate -0.8

2 Show that $z = 3$ is a root of the cubic equation $z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0$ and find the other roots.

OCR MEI FP1 2013 June Q2

6 marks Moderate -0.5

2 You are given that $z = \frac { 3 } { 2 }$ is a root of the cubic equation $2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0$. Find the other two roots.

AQA Further AS Paper 1 2018 June Q8

5 marks Standard +0.8

8

Find the other roots.
Determine the value of $m$.

WJEC Further Unit 1 2019 June Q5

6 marks Moderate -0.5

5. Given that $x = - \frac { 1 } { 2 }$ and $x = - 3$ are two roots of the equation $$2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 23 x + 15 = 0$$ find the remaining roots.

CAIE P3 2017 March Q8

10 marks Standard +0.3

Showing all your working, verify that $u$ is a root of the equation $\mathrm { p } ( z ) = 0$.
Find the other three roots of the equation $\mathrm { p } ( z ) = 0$.