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 \(K\).

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]

Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x^3 - 4 \equiv (x - 1)(Ax^2 + Bx + C) + D.$$ [5]