Question 2 - A-Level Maths

Exam Board	AQA
Module	C4 (Core Mathematics 4)
Year	2011
Session	January
Topic	Factor & Remainder Theorem
Type	Verify factor then simplify rational expression

The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).