Expand from factored form

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)

1. The curve \(C\) has equation

$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$

1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
3. Find the \(x\)-coordinates of these two points.

2 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

4

1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.

6 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

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

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.