Find constants using remainder theorem - Factor & Remainder Theorem

CAIE P2 2011 November Q5

5 The polynomial $4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9$, where $a$ is a constant, is denoted by $\mathrm { p } ( x )$. It is given that when $\mathrm { p } ( x )$ is divided by $( 2 x - 1 )$ the remainder is 10 .

Find the value of $a$ and hence verify that ( $x - 3$ ) is a factor of $\mathrm { p } ( x )$.
When $a$ has this value, solve the equation $\mathrm { p } ( x ) = 0$.

CAIE P2 2016 November Q4

4 The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where $a$ is a constant.

Use the factor theorem to show that ( $x + 1$ ) is a factor of $\mathrm { p } ( x )$ for all values of $a$.
Given that the remainder is - 42 when $\mathrm { p } ( x )$ is divided by ( $x - 2$ ), find the value of $a$.
When $a$ has the value found in part (ii), factorise $\mathrm { p } \left( x ^ { 2 } \right)$ completely.

Edexcel P2 2024 June Q4

4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where $k$ is a constant.

State the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ) Given that ( $2 x - 1$ ) is a factor of $\mathrm { f } ( x )$
show that $k = 11$
Hence
1. fully factorise $\mathrm { f } ( x )$,
2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.

Edexcel P2 2019 October Q4

4. $\mathrm { f } ( x ) = ( x - 3 ) \left( 3 x ^ { 2 } + x + a \right) - 35$ where $a$ is a constant

State the remainder when $\mathrm { f } ( x )$ is divided by $( x - 3 )$. Given $( 3 x - 2 )$ is a factor of $\mathrm { f } ( x )$,
show that $a = - 17$
Using algebra and showing each step of your working, fully factorise $\mathrm { f } ( x )$.

OCR C2 2005 June Q5

5 The cubic polynomial $\mathrm { f } ( x )$ is given by $$f ( x ) = x ^ { 3 } + a x + b$$ where $a$ and $b$ are constants. It is given that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$ and that the remainder when $\mathrm { f } ( x )$ is divided by $( x - 3 )$ is 16 .

Find the values of $a$ and $b$.
Hence verify that $\mathrm { f } ( 2 ) = 0$, and factorise $\mathrm { f } ( x )$ completely.

AQA C4 2015 June Q3

4 marks

3

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d$, where $d$ is a constant. When $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ), the remainder is - 2 . Use the Remainder Theorem to find the value of $d$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3$.
1. Given that $x = - \frac { 1 } { 2 }$ is a solution of the equation $\mathrm { g } ( x ) = 0$, write $\mathrm { g } ( x )$ as a product of three linear factors.
2. The function h is defined by $\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }$ for $x > 2$. Simplify $\mathrm { h } ( x )$, and hence show that h is a decreasing function.
  [0pt] [4 marks]

Edexcel C2 Q5

5. $\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 1 .
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is 28 .

Find the value of $a$ and the value of $b$.
Show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.