Find constant, factorise completely

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + k x - 30$$ where \(k\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(k\).
2. Hence find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) and factorise \(\mathrm { p } ( x )\) completely.
3. It is given that \(a\) is one of the roots of the equation \(\mathrm { p } ( x ) = 0\). Given also that the equation \(| 4 y - 5 | = a\) is satisfied by two real values of \(y\), find these two values of \(y\).

2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).

1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value,
   (a) factorise \(\mathrm { p } ( x )\) completely,
   (b) find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

2

1. Given that \(( x + 2 )\) is a factor of $$4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18$$ find the value of the constant \(a\).
2. When \(a\) has this value, factorise \(4 x ^ { 3 } + a x ^ { 2 } - ( a + 1 ) x - 18\) completely.

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + 14 x + a + 1$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to find the value of \(a\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Hence, without using a calculator, solve the equation \(\mathrm { f } ( 2 x ) = 3 \mathrm { f } ( x )\).

4 The polynomial \(a x ^ { 3 } - 20 x ^ { 2 } + x + 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value,
   (a) factorise \(\mathrm { p } ( x )\),
   (b) solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\). When \(a\) has this value,
2. factorise \(\mathrm { p } ( x )\),
3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

3 The polynomial \(x ^ { 4 } - 6 x ^ { 2 } + x + a\) is denoted by \(\mathrm { f } ( x )\).

1. It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\). Find the value of \(a\).
2. When \(a\) has this value, verify that ( \(x - 2\) ) is also a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.

2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

3. $$f ( x ) = ( 3 x - 2 ) ( x - k ) - 8$$ where \(k\) is a constant.

1. Write down the value of \(\mathrm { f } ( k )\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 4
2. Find the value of \(k\).
3. Factorise f(x) completely.

3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that \(a = - 9\)
2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.

4.
\(\mathrm { f } ( x ) = - 4 x ^ { 3 } + a x ^ { 2 } + 9 x - 18\), where \(a\) is a constant. Given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),

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

1. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + p x + 6$$ where \(p\) is a constant.
Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find

1. the value of \(p\),
2. the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).

2. \(f ( x ) = x ^ { 3 } + k x - 20\). Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),

1. find the value of the constant \(k\),
2. solve the equation \(\mathrm { f } ( x ) = 0\).

6. 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.

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$

2 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2\), where \(a\) is a constant. You are given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Find all the factors of \(\mathrm { f } ( x )\).

2. $$f ( x ) = x ^ { 3 } + k x - 20 .$$ Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),

1. find the value of the constant \(k\),
2. solve the equation \(\mathrm { f } ( x ) = 0\).

1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.

1. 1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
   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 \(x - 3\).
3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)

2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant. Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
3. Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).