OCR MEI C1 (Core Mathematics 1)

1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$

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 \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

3

1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
   (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
   (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
   Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.

4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).