Question 5 - A-Level Maths

Exam Board	AQA
Module	C1 (Core Mathematics 1)
Year	2011
Session	January
Topic	Factor & Remainder Theorem
Type	Direct remainder then factorise

1. Sketch the curve with equation $y = x ( x - 2 ) ^ { 2 }$.
2. Show that the equation $x ( x - 2 ) ^ { 2 } = 3$ can be expressed as $$x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3 = 0$$
The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x - 3$.
1. Find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
2. Use the Factor Theorem to show that $x - 3$ is a factor of $\mathrm { p } ( x )$.
3. Express $\mathrm { p } ( x )$ in the form $( x - 3 ) \left( x ^ { 2 } + b x + c \right)$, where $b$ and $c$ are integers.
Hence show that the equation $x ( x - 2 ) ^ { 2 } = 3$ has only one real root and state the value of this root.