Two unknowns with show-that step - Factor & Remainder Theorem

Edexcel C12 2016 June Q7

7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that $( x - 2 )$ is a factor of $\mathrm { f } ( x )$,

use the factor theorem to show that $2 a + b = - 7$ Given also that when $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is - 36
find the value of $a$ and the value of $b$.
$\mathrm { f } ( x )$ can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
1. Find $\mathrm { Q } ( x )$.
2. Prove that the equation $\mathrm { f } ( x ) = 0$ has only one real root. You must justify your answer and show all your working.

Edexcel C12 2018 June Q2

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

Show that $$a + 4 b = 28$$ When $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is - 17
Find the value of $a$ and the value of $b$.

Edexcel C12 2018 October Q8

8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where $p$ and $q$ are constants.
When $\mathrm { f } ( x )$ is divided by $( x - 1 )$, the remainder is - 6

Use the remainder theorem to show that $p + q = - 5$ Given also that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $p$ and the value of $q$.
Factorise $\mathrm { f } ( \mathrm { x } )$ completely.

Edexcel C12 Specimen Q6

6. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 1 )$, the remainder is 7

Show that $a + b = 3$ When $\mathrm { f } ( x )$ is divided by ( $x + 2$ ), the remainder is - 8
Find the value of $a$ and the value of $b$.

Edexcel P2 2021 January Q1

1. $$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is 4

Show that $a + b = - 1$ When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is - 23
Find the value of $a$ and the value of $b$.

Edexcel P2 2018 Specimen Q1

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x - 1$ ), the remainder is 7

Show that $a + b = 3$ When $\mathrm { f } ( x )$ is divided by ( $x + 2$ ), the remainder is - 8
Find the value of $a$ and the value of $b$

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Edexcel C2 2011 January Q1

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 1 )$, the remainder is 7 .

Show that $a + b = 3$. When $\mathrm { f } ( x )$ is divided by $( x + 2 )$, the remainder is - 8 .
Find the value of $a$ and the value of $b$.

Edexcel C2 2012 January Q5

$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 3$, where $a$ and $b$ are constants.

Given that when $\mathrm { f } ( x )$ is divided by $( x + 2 )$ the remainder is 7 ,

show that $2 a - b = 6$ Given also that when $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is 4 ,
find the value of $a$ and the value of $b$.

Edexcel C2 2015 June Q3

3. $\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B$, where $A$ and $B$ are constants. Given that when $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is 45 ,

show that $B - A = 48$ Given also that ( $2 x + 1$ ) is a factor of $\mathrm { f } ( x )$,
find the value of $A$ and the value of $B$.
Factorise f(x) fully.

Edexcel C2 2018 June Q3

3. $$f ( x ) = 24 x ^ { 3 } + A x ^ { 2 } - 3 x + B$$ where $A$ and $B$ are constants.
When $\mathrm { f } ( x )$ is divided by $( 2 x - 1 )$ the remainder is 30

Show that $A + 4 B = 114$ Given also that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$,
find another equation in $A$ and $B$.
Find the value of $A$ and the value of $B$.
Hence find a quadratic factor of $\mathrm { f } ( x )$.

OCR C2 Q8

8. The polynomial $\mathrm { p } ( x )$ is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$$ where $a$ and $b$ are constants.
Given that when $\mathrm { p } ( x )$ is divided by $( x + 2 )$ there is a remainder of 20 ,

find an expression for $b$ in terms of $a$. Given also that $( 2 x - 1 )$ is a factor of $\mathrm { p } ( x )$,
find the values of $a$ and $b$,
fully factorise $\mathrm { p } ( x )$.