Direct remainder then factorise

7 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) the remainder is 30 , and that when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) the remainder is 18 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, verify that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\) and hence factorise \(\mathrm { p } ( x )\) completely.

3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
   1. \(x - 2\)
   2. \(x + 3\)
2. Hence factorise \(\mathrm { f } ( x )\) completely.

5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ).
2. Use the factor theorem to show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
4. Use your answer to part (c) and the sketch to deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)

4. $$f ( x ) = 6 x ^ { 3 } - 7 x ^ { 2 } - 43 x + 30$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
   1. \(2 x + 1\)
   2. \(x - 3\)
2. Hence factorise \(\mathrm { f } ( x )\) completely.

2. $$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 58 x + 40$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ). Given that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
2. find all the solutions of \(\mathrm { f } ( x ) = 0\).

1. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 5 x + 4$$

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

4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
3. Factorise \(\mathrm { f } ( x )\) completely.

$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
3. Factorise \(\mathrm { f } ( x )\) completely.

$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. factorise \(\mathrm { f } ( x )\) completely.

8 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 33\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
2. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form as simply as possible. On its first trip between Malby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses \(2 \%\) more coal than the previous trip.

7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

5

1. 1. Sketch the curve with equation \(y = x ( x - 2 ) ^ { 2 }\).
   2. Show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
   2. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x - 3 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
3. Hence show that the equation \(x ( x - 2 ) ^ { 2 } = 3\) has only one real root and state the value of this root.

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
2. 1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of linear factors.
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\), stating the values of \(x\) where the curve meets the \(x\)-axis.

3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$$

1. 1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
3. 1. Verify that \(\mathrm { p } ( - 1 ) < \mathrm { p } ( 0 )\).
   2. Sketch the curve with equation \(y = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15\), indicating the values where the curve crosses the coordinate axes.

5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
2. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
3. 1. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   2. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

7

1. Sketch the curve with equation \(y = x ^ { 2 } ( x - 3 )\).
2. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 4\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x )\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root and state its value.
      [0pt] [3 marks]

4 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 48\).

1. 1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as a product of three linear factors.
2. 1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
   2. Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + b x + c \right) + r\), where \(b , c\) and \(r\) are integers. [3 marks]

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. 1. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
3. 1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
   2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.

4

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below.
   \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
   1. State the \(y\)-coordinate of the point \(B\).
   2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
2. 1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$