Apply remainder theorem only - Factor & Remainder Theorem

9 marks Moderate -0.8

6. $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1$$

Find the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x - 1$ ).
1. Find the remainder when $\mathrm { f } ( x )$ is divided by $( x + 2 )$.
2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0$$

7 marks Moderate -0.3

1 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7$.

Use the Remainder Theorem to find the remainder when $\mathrm { f } ( x )$ is divided by $( 2 x + 1 )$.
(2 marks)
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = \mathrm { f } ( x ) + d$, where $d$ is a constant.
1. Given that $( 2 x + 1 )$ is a factor of $\mathrm { g } ( x )$, show that $\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4$.
  (1 mark)
2. Given that $\mathrm { g } ( x )$ can be written as $\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)$, where $a$ is an integer, express $\mathrm { g } ( x )$ as a product of three linear factors.
3. Hence, or otherwise, show that $\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }$, where $p$ and $q$ are integers.
  (3 marks)

3 marks Moderate -0.8

$$f(x) = 4x^3 + 3x^2 - 2x - 6.$$ Find the remainder when $f(x)$ is divided by $(2x + 1)$. [3]

3 marks Moderate -0.8

When $x^3 - kx + 4$ is divided by $x - 3$, the remainder is 1. Use the remainder theorem to find the value of $k$. [3]

3 marks Moderate -0.8

You are given that $\text{f}(x) = x^5 + kx - 20$. When $\text{f}(x)$ is divided by $(x - 2)$, the remainder is 18. Find the value of $k$. [3]

3 marks Moderate -0.8

$f(x) = 4x^3 + 3x^2 - 2x - 6$. Find the remainder when $f(x)$ is divided by $(2x + 1)$. [3]

3 marks Moderate -0.8

Find the remainder when $f(x) = 4x^3 + 3x^2 - 2x - 6$ is divided by $(2x + 1)$. [3]