One factor, one non-zero remainder - Factor & Remainder Theorem

CAIE P2 2021 March Q6

Moderate -0.3

6 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x + b$$ where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is 5 when $\mathrm { p } ( x )$ is divided by $( x - 3 )$.

Find the values of $a$ and $b$.
Hence find the exact root of the equation $\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0$.

CAIE P2 2020 November Q2

Moderate -0.8

2 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is 72 when $\mathrm { p } ( x )$ is divided by $( x - 2 )$. Find the values of $a$ and $b$.

CAIE P2 2021 November Q1

Moderate -0.8

1 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x - 10$$ where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is - 55 when $\mathrm { p } ( x )$ is divided by $( x + 3 )$.

Find the values of $a$ and $b$.
Hence factorise $\mathrm { p } ( x )$ completely.

CAIE P2 2023 November Q5

Moderate -0.3

5 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 20$$ where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is - 11 when $\mathrm { p } ( x )$ is divided by $( x + 1 )$.

Find the values of $a$ and $b$.
Hence factorise $\mathrm { p } ( x )$, and determine the exact roots of the equation $\mathrm { p } ( 3 x ) = 0$.

CAIE P2 2024 November Q5

Standard +0.3

5 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where $a$ and $b$ are constants.It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ ,and that the remainder is 24 when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ .

Find the values of $a$ and $b$ .
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
Factorise $\mathrm { p } ( x )$ and hence show that the equation $\mathrm { p } ( x ) = 0$ has exactly one real root.
Solve the equation $\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0$ for $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations $y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }$ and $y = \frac { 27 } { 2 x + 5 }$ for $x \geqslant 0$.
The curves meet at the point $( 2,3 )$.
Region $A$ is bounded by the curve $y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }$ and the straight lines $x = 0 , x = 2$ and $y = 0$.
Region $B$ is bounded by the two curves and the straight line $x = 0$.

CAIE P2 2005 June Q4

Moderate -0.8

4 The polynomial $x ^ { 3 } - x ^ { 2 } + a x + b$ is denoted by $\mathrm { p } ( x )$. It is given that ( $x + 1$ ) is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 12 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$.

CAIE P2 2007 June Q4

Moderate -0.3

4 The polynomial $2 x ^ { 3 } - 3 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x - 2 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is - 20 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, find the remainder when $\mathrm { p } ( x )$ is divided by ( $x ^ { 2 } - 4$ ).

CAIE P2 2008 June Q4

Moderate -0.8

4 The polynomial $2 x ^ { 3 } + 7 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x + 1 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is 5 . Find the values of $a$ and $b$.

CAIE P2 2009 June Q6

Moderate -0.8

6 The polynomial $x ^ { 3 } + a x ^ { 2 } + b x + 6$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x - 2 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x - 1 )$ the remainder is 4 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, find the other two linear factors of $\mathrm { p } ( x )$.

CAIE P2 2011 June Q7

Moderate -0.3

7 The cubic polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that, when $\mathrm { p } ( x )$ is divided by ( $x + 1$ ), the remainder is 24 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.

CAIE P2 2013 June Q4

Moderate -0.3

4 The polynomial $a x ^ { 3 } - 5 x ^ { 2 } + b x + 9$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 3 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x + 1 )$ the remainder is 8 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.

CAIE P3 2010 June Q5

Moderate -0.8

5 The polynomial $2 x ^ { 3 } + 5 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is 9 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.

CAIE P3 2011 June Q5

Moderate -0.3

5 The polynomial $a x ^ { 3 } + b x ^ { 2 } + 5 x - 2$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x - 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 12 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, find the quadratic factor of $\mathrm { p } ( x )$.

CAIE P3 2013 June Q5

Moderate -0.3

5 The polynomial $8 x ^ { 3 } + a x ^ { 2 } + b x + 3$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by ( $2 x - 1$ ) the remainder is 1 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, find the remainder when $\mathrm { p } ( x )$ is divided by $2 x ^ { 2 } - 1$.

CAIE P3 2014 November Q3

Moderate -0.5

3 The polynomial $a x ^ { 3 } + b x ^ { 2 } + x + 3$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 3 x + 1 )$ is a factor of $\mathrm { p } ( x )$, and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 21 . Find the values of $a$ and $b$.

CAIE P3 2015 November Q6

Moderate -0.3

6 The polynomial $8 x ^ { 3 } + a x ^ { 2 } + b x - 1$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x + 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( 2 x + 1 )$ the remainder is 1 .

Find the values of $a$ and $b$.
When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.

CAIE P3 2019 November Q2

Moderate -0.8

2 The polynomial $6 x ^ { 3 } + a x ^ { 2 } + b x - 2$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x + 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is - 24 . Find the values of $a$ and $b$.

Edexcel P2 2020 October Q3

Moderate -0.3

3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by ( $x + 4$ ), the remainder is - 108

Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( $2 x - 1$ ) is a factor of $\mathrm { f } ( x )$,
find the value of $a$ and the value of $b$.
Find $\mathrm { f } ^ { \prime } ( x )$.
Hence find the exact coordinates of the stationary points of the curve with equation $y = \mathrm { f } ( x )$.
VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P2 2023 October Q4

Moderate -0.3

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

Solutions relying on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + a x ^ { 2 } - 29 x + b$$ where $a$ and $b$ are constants.
Given that $( 2 x + 1 )$ is a factor of $\mathrm { f } ( x )$,

show that $$a + 4 b = - 56$$ Given also that when $\mathrm { f } ( x )$ is divided by $( x - 2 )$ the remainder is - 25
find a second simplified equation linking $a$ and $b$.
Hence, using algebra and showing your working,
1. find the value of $a$ and the value of $b$,
2. fully factorise $\mathrm { f } ( x )$.

Edexcel C2 2010 January Q3

Moderate -0.3

3. $$f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x - 6$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by $( 2 x - 1 )$ the remainder is - 5 .
When $\mathrm { f } ( x )$ is divided by $( x + 2 )$ there is no remainder.

Find the value of $a$ and the value of $b$.
Factorise $\mathrm { f } ( x )$ completely.

Edexcel C2 2013 January Q2

Moderate -0.8

2. $\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 4 x - 3$, where $a$ and $b$ are constants. Given that $( x - 1 )$ is a factor of $\mathrm { f } ( x )$,

show that $$a + b = 7$$ Given also that, when $\mathrm { f } ( x )$ is divided by $( x + 2 )$, the remainder is 9 ,
find the value of $a$ and the value of $b$, showing each step in your working.

Edexcel C2 2014 January Q2

Moderate -0.3

2. $\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$, where $a$ and $b$ are constants. Given that when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ) the remainder is 25 ,

show that $2 a + b = 5$ Given also that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $a$ and the value of $b$.
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\includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}

AQA C1 2012 January Q5

Moderate -0.8

5 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x - 12$, where $c$ and $d$ are constants.

When $\mathrm { p } ( x )$ is divided by $x + 2$, the remainder is - 150 . Show that $2 c - d + 65 = 0$.
Given that $x - 3$ is a factor of $\mathrm { p } ( x )$, find another equation involving $c$ and $d$.
By solving these two equations, find the value of $c$ and the value of $d$.

AQA C1 2014 June Q5

3 marks Moderate -0.8

5 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x + 3$$ where $c$ and $d$ are integers.

Given that $x + 3$ is a factor of $\mathrm { p } ( x )$, show that $$3 c - d = 8$$
The remainder when $\mathrm { p } ( x )$ is divided by $x - 2$ is 65 . Obtain a further equation in $c$ and $d$.
Use the equations from parts (a) and (b) to find the value of $c$ and the value of $d$. [3 marks]

Edexcel C2 Q1

Moderate -0.8

1. $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14 , \text { where } a \text { and } b \text { are constants. }$$ Given that when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) the remainder is 9 ,

write down an equation connecting $a$ and $b$. Given also that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
find the values of $a$ and $b$.