Polynomial identity or expansion

9. \(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant (a) Write down the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + k )\). When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 15
(b) Show that \(k = 2\)
(c) Hence factorise \(\mathrm { f } ( x )\) completely. \section*{9.} " .
\(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant

1. Given that

$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Given that \(z\) is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$

2 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x ^ { 3 } - 3 \equiv ( x + 3 ) \left( A x ^ { 2 } + B x + C \right) + D\).

1 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x \left( x ^ { 2 } - 5 \right) \equiv ( x - 2 ) \left( A x ^ { 2 } + B x + C \right) + D\).

8 The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 .

1. Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
2. Prove that \(K\) has at least two distinct factors other than 1 and \(K\).

2. $$f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9 , \text { where } p \text { is a constant. }$$

1. Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
2. Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
3. Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).

3. \(f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9\), where \(p\) is a constant.

1. Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
2. Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
3. Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).