Polynomial Division with Remainder Verification - Polynomial Division & Manipulation

CAIE P2 2010 June Q4

6 marks Moderate -0.8

4 The polynomial $x ^ { 3 } + 3 x ^ { 2 } + 4 x + 2$ is denoted by $\mathrm { f } ( x )$.

Find the quotient and remainder when $\mathrm { f } ( x )$ is divided by $x ^ { 2 } + x - 1$.
Use the factor theorem to show that $( x + 1 )$ is a factor of $\mathrm { f } ( x )$.

CAIE P3 2013 June Q1

3 marks Easy -1.8

1 Find the quotient and remainder when $2 x ^ { 2 }$ is divided by $x + 2$.

CAIE P3 2017 November Q1

3 marks Moderate -0.8

1 Find the quotient and remainder when $x ^ { 4 }$ is divided by $x ^ { 2 } + 2 x - 1$.

CAIE P2 2016 March Q1

3 marks Easy -1.2

1 Find the quotient and the remainder when $2 x ^ { 3 } + 3 x ^ { 2 } + 10$ is divided by $( x + 2 )$.

CAIE P2 2005 November Q2

6 marks Moderate -0.3

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

Find the remainder when $\mathrm { p } ( x )$ is divided by $x - 1$.
Find the quotient and remainder when $\mathrm { p } ( x )$ is divided by $x ^ { 2 } + x - 1$.

CAIE P2 2019 November Q1

3 marks Moderate -0.5

1 The polynomial $\mathrm { f } ( x )$ is defined by $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - 6 x + 11$$ Find the quotient and remainder when $\mathrm { f } ( x )$ is divided by $\left( x ^ { 2 } + 2 \right)$.

CAIE P3 2020 June Q1

3 marks Moderate -0.5

1 Find the quotient and remainder when $6 x ^ { 4 } + x ^ { 3 } - x ^ { 2 } + 5 x - 6$ is divided by $2 x ^ { 2 } - x + 1$.

CAIE P3 2023 June Q2

3 marks Moderate -0.5

2 Find the quotient and remainder when $2 x ^ { 4 } - 27$ is divided by $x ^ { 2 } + x + 3$.

CAIE P3 2024 March Q1

3 marks Moderate -0.5

1 Find the quotient and remainder when $x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 12 x + 27$ is divided by $x ^ { 2 } + 5$.

CAIE P3 2021 November Q1

3 marks Moderate -0.5

1 Find the quotient and remainder when $2 x ^ { 4 } + 1$ is divided by $x ^ { 2 } - x + 2$.

OCR C4 2005 June Q1

4 marks Moderate -0.3

1 Find the quotient and the remainder when $x ^ { 4 } + 3 x ^ { 3 } + 5 x ^ { 2 } + 4 x - 1$ is divided by $x ^ { 2 } + x + 1$.

OCR C4 2008 June Q1

6 marks Moderate -0.3

1

Simplify $\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }$.
Find the quotient and remainder when $x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5$ is divided by $x ^ { 2 } + 4 x + 1$.

OCR C4 Specimen Q1

4 marks Moderate -0.8

1 Find the quotient and remainder when $x ^ { 4 } + 1$ is divided by $x ^ { 2 } + 1$.

OCR C4 Q2

6 marks Moderate -0.3

(i) Simplify

$$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$ (ii) Find the quotient and remainder when ( $2 x ^ { 4 } - 1$ ) is divided by ( $x ^ { 2 } - 2$ ).

OCR C4 2010 January Q1

4 marks Moderate -0.3

1 Find the quotient and the remainder when $x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1$ is divided by $x ^ { 2 } + 5 x + 2$.

OCR C4 2009 June Q1

4 marks Moderate -0.3

1 Find the quotient and the remainder when $3 x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } - 14 x - 8$ is divided by $x ^ { 2 } + x + 2$.

OCR C4 2016 June Q1

3 marks Moderate -0.5

1 Find the quotient and the remainder when $4 x ^ { 3 } + 8 x ^ { 2 } - 5 x + 12$ is divided by $2 x ^ { 2 } + 1$.

AQA C4 2005 June Q3

6 marks Moderate -0.8

3

Find the remainder when $2 x ^ { 3 } - x ^ { 2 } + 2 x - 2$ is divided by $2 x - 1$.
Given that $\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }$, find the values of $a$ and $b$.

AQA C4 2009 June Q1

5 marks Moderate -0.8

1

Use the Remainder Theorem to find the remainder when $3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5$ is divided by $3 x - 1$.
Express $\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }$ in the form $a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }$, where $a , b$ and $c$ are integers.