Polynomial with equal remainders - Factor & Remainder Theorem

CAIE P2 2010 November Q3

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

Find the value of $a$.
When $a$ has this value, show that $( x - 1 )$ is a factor of $\mathrm { p } ( x )$ and find the quotient when $\mathrm { p } ( x )$ is divided by $( x - 1 )$.

CAIE P3 2024 November Q1

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

Edexcel P2 2022 January Q5

5. $$f ( x ) = 3 x ^ { 3 } + A x ^ { 2 } + B x - 10$$ where $A$ and $B$ are integers.
Given that

when $\mathrm { f } ( x )$ is divided by $( x - 1 )$ the remainder is $k$
when $\mathrm { f } ( x )$ is divided by $( x + 1 )$ the remainder is $- 10 k$
$k$ is a constant
1. show that

$$11 A + 9 B = 83$$ Given also that $( 3 x - 2 )$ is a factor of $\mathrm { f } ( x )$,

find the value of $A$ and the value of $B$.

Hence find the quadratic expression $\mathrm { g } ( x )$ such that $$f ( x ) = ( 3 x - 2 ) g ( x )$$

Edexcel C2 2009 January Q6

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

Find the value of $a$. Given that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $b$.

OCR MEI C1 Q7

7 The remainder when $x ^ { 3 } - 2 x + 4$ is divided by ( $x - 2$ ) is twice the remainder when $x ^ { 2 } + x + k$ is divided by ( $x + 1$ ).
Find the value of $k$.

OCR C2 Specimen Q9

9 The cubic polynomial $x ^ { 3 } + a x ^ { 2 } + b x - 6$ is denoted by $\mathrm { f } ( x )$.

The remainder when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ) is equal to the remainder when $\mathrm { f } ( x )$ is divided by $( x + 2 )$. Show that $b = - 4$.
Given also that ( $x - 1$ ) is a factor of $\mathrm { f } ( x )$, find the value of $a$.
With these values of $a$ and $b$, express $\mathrm { f } ( x )$ as a product of a linear factor and a quadratic factor.
Hence determine the number of real roots of the equation $\mathrm { f } ( x ) = 0$, explaining your reasoning.

OCR C2 Q9

9. The polynomial $\mathrm { f } ( x )$ is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where $k$ is a constant.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ) the remainder is $r$.
When $\mathrm { f } ( x )$ is divided by $( x - 3 )$ the remainder is $3 r$.

Find the value of $k$.
Find the value of $r$.
Show that $( x - 5 )$ is a factor of $\mathrm { f } ( x )$.
Show that there is only one real solution to the equation $\mathrm { f } ( x ) = 0$.

Edexcel C2 Q1

1. $$f ( x ) = p x ^ { 3 } + 6 x ^ { 2 } + 12 x + q .$$ Given that the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) is equal to the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ),

find the value of $p$. Given also that $q = 3$, and $p$ has the value found in part (a),
find the value of the remainder.

Edexcel C2 Q9

9. The polynomial $\mathrm { f } ( x )$ is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where $k$ is a constant.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ) the remainder is $r$.
When $\mathrm { f } ( x )$ is divided by $( x - 3 )$ the remainder is $3 r$.

Find the value of $k$.
Find the value of $r$.
Show that $( x - 5 )$ is a factor of $\mathrm { f } ( x )$.
Show that there is only one real solution to the equation $\mathrm { f } ( x ) = 0$. END