Factorise then sketch

1. (a) Factorise completely \(x ^ { 3 } - 4 x\)
   (b) Sketch the curve \(C\) with equation

$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.

9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),

1. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
2. Hence factorise \(\mathrm { f } ( x )\) completely.
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.

1. (a) Factorise completely \(9 x - 4 x ^ { 3 }\)
   (b) Sketch the curve \(C\) with equation

$$y = 9 x - 4 x ^ { 3 }$$ Show on your sketch the coordinates at which the curve meets the \(x\)-axis. The points \(A\) and \(B\) lie on \(C\) and have \(x\) coordinates of - 2 and 1 respectively.
(c) Show that the length of \(A B\) is \(k \sqrt { } 10\) where \(k\) is a constant to be found.

4. $$f ( x ) = 4 x - 3 x ^ { 2 } - x ^ { 3 }$$

1. Fully factorise \(4 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.