One factor, one non-zero remainder

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 5 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).

1. Find the values of \(a\) and \(b\).
2. Hence find the exact root of the equation \(\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0\).

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).

1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x - 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 55 when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 20$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is - 11 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\), and determine the exact roots of the equation \(\mathrm { p } ( 3 x ) = 0\).

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) .
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

4 The polynomial \(x ^ { 3 } - x ^ { 2 } + a x + b\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\).

4 The polynomial \(2 x ^ { 3 } - 3 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 20 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by ( \(x ^ { 2 } - 4\) ).

4 The polynomial \(2 x ^ { 3 } + 7 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).

6 The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the other two linear factors of \(\mathrm { p } ( x )\).

7 The cubic polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that, when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ), the remainder is 24 .

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

4 The polynomial \(a x ^ { 3 } - 5 x ^ { 2 } + b x + 9\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 3 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 8 .

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

5 The polynomial \(2 x ^ { 3 } + 5 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 9 .

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

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).

5 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(2 x ^ { 2 } - 1\).

3 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 21 . Find the values of \(a\) and \(b\).

6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 1 )\) the remainder is 1 .

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

2 The polynomial \(6 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 24 . Find the values of \(a\) and \(b\).

3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 4\) ), the remainder is - 108

1. Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the exact coordinates of the stationary points of the curve with equation \(y = \mathrm { f } ( x )\).

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + a x ^ { 2 } - 29 x + b$$ where \(a\) and \(b\) are constants.
Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a + 4 b = - 56$$ Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 25
2. find a second simplified equation linking \(a\) and \(b\).
3. Hence, using algebra and showing your working,
   1. find the value of \(a\) and the value of \(b\),
   2. fully factorise \(\mathrm { f } ( x )\).

3. $$f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x - 6$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is - 5 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is no remainder.

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

2. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 4 x - 3\), where \(a\) and \(b\) are constants. Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a + b = 7$$ Given also that, when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is 9 ,
2. find the value of \(a\) and the value of \(b\), showing each step in your working.

2. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) the remainder is 25 ,

1. show that \(2 a + b = 5\) Given also that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\).
   \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_90_97_2613_1784}
   \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}

5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x - 12\), where \(c\) and \(d\) are constants.

1. When \(\mathrm { p } ( x )\) is divided by \(x + 2\), the remainder is - 150 . Show that \(2 c - d + 65 = 0\).
2. Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find another equation involving \(c\) and \(d\).
3. By solving these two equations, find the value of \(c\) and the value of \(d\).

5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x + 3$$ where \(c\) and \(d\) are integers.

1. Given that \(x + 3\) is a factor of \(\mathrm { p } ( x )\), show that $$3 c - d = 8$$
2. The remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\) is 65 . Obtain a further equation in \(c\) and \(d\).
3. Use the equations from parts (a) and (b) to find the value of \(c\) and the value of \(d\). [3 marks]

1. $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14 , \text { where } a \text { and } b \text { are constants. }$$ Given that when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ) the remainder is 9 ,

1. write down an equation connecting \(a\) and \(b\). Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the values of \(a\) and \(b\).