Remainder condition

1. $$f ( x ) = a x ^ { 3 } + 3 x ^ { 2 } - 8 x + 2 \quad \text { where } a \text { is a constant }$$ Given that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 3 , find the value of \(a\).

6 When \(x ^ { 3 } + 3 x + k\) is divided by \(x - 1\), the remainder is 6 . Find the value of \(k\). \begin{figure}[h] \end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C . Find the equation of the line BC and the \(x\)-coordinate of C .

6 When \(x ^ { 3 } + k x + 7\) is divided by \(( x - 2 )\), the remainder is 3 . Find the value of \(k\).

11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).

1. You are asked to find the values of \(k\) which satisfy the following conditions.
   (A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
   (B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
   (C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
   (D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
   (E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
2. Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\). Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.

8 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).
\(9 n\) is a positive integer. Show that \(n ^ { 2 } + n\) is always even.

1. \(f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 3 x - 2 )\).

4 When \(x ^ { 3 } + k x + 5\) is divided by \(x - 2\), the remainder is 3 . Use the remainder theorem to find the value of \(k\).

7 When \(x ^ { 3 } + 2 x ^ { 2 } + 5 x + k\) is divided by ( \(x + 3\) ), the remainder is 6 . Find the value of \(k\).

3 When \(x ^ { 3 } - k x + 4\) is divided by \(x - 3\), the remainder is 1 . Use the remainder theorem to find the value of \(k\).

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

8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).

2. $$f ( x ) = ( 2 x - 3 ) ( x - k ) - 12$$ where \(k\) is a constant.
a.Write down the value of \(\mathrm { f } ( k )\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) the remainder is - 5
b. find the value of \(k\).
c. Factorise \(\mathrm { f } ( x )\) completely.

1. \(\quad \mathrm { f } ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).

Given that when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) there is a remainder of - 35 ,

1. find the value of the constant \(k\),
2. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(3 x - 2\) ).

5. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6 ,$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),

1. find the value of the constant \(p\),
2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.