Two factors given

2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 9 x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) and \(( 2 x + 1 )\) are factors of \(\mathrm { p } ( x )\).
Find the values of \(a\) and \(b\).

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

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

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

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 \(\left( x ^ { 2 } + 1 \right)\).

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 \(( x + 1 )\) and \(( x - 2 )\) are factors of \(\mathrm { p } ( x )\).

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

4

1. The polynomial \(a x ^ { 3 } + b x ^ { 2 } - 25 x - 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 3 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.

5

1. Given that ( \(x + 2\) ) and ( \(x + 3\) ) are factors of $$5 x ^ { 3 } + a x ^ { 2 } + b$$ find the values of the constants \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, factorise $$5 x ^ { 3 } + a x ^ { 2 } + b$$ completely, and hence solve the equation $$5 ^ { 3 y + 1 } + a \times 5 ^ { 2 y } + b = 0$$ giving any answers correct to 3 significant figures.

4 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + k x + c\). The value of \(\mathrm { f } ( 0 )\) is 6, and \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the values of \(k\) and \(c\).

8 Two cubic polynomials are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + ( a - 3 ) x + 2 b , \quad \mathrm {~g} ( x ) = 3 x ^ { 3 } + x ^ { 2 } + 5 a x + 4 b$$ where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have a common factor of ( \(x - 2\) ), show that \(a = - 4\) and find the value of \(b\).
2. Using these values of \(a\) and \(b\), factorise \(\mathrm { f } ( x )\) fully. Hence show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) have two common factors.

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where \(a\) and \(b\) are constants.
Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
2. express \(\mathrm { f } ( x )\) in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where \(k\) is a constant to be found.
3. Hence,
   1. state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
   2. write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)

3 It is given that \(( x + 1 )\) and \(( x - 3 )\) are two factors of \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = p x ^ { 3 } - 3 x ^ { 2 } - 8 x + q$$ 3

1. Find the values of \(p\) and \(q\).
   [0pt] [3 marks]
   3
2. Fully factorise f (x).
   \section*{Fully justify your answer.}