Verify factor then sketch or analyse curve

6 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$

1. 1. Using the factor theorem, show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Hence express \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Sketch the curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 10 x + 8\), showing the coordinates of the points where the curve cuts the axes.
   (You are not required to calculate the coordinates of the stationary points.)

9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).

1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Fully factorise \(\mathrm { f } ( x )\).
3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).

A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).

1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]

You are given that \(\text{f}(x) = x^3 + 9x^2 + 20x + 12\).

1. Show that \(x = -2\) is a root of \(\text{f}(x) = 0\). [2]
2. Divide \(\text{f}(x)\) by \(x + 6\). [2]
3. Express \(\text{f}(x)\) in fully factorised form. [2]
4. Sketch the graph of \(y = \text{f}(x)\). [3]
5. Solve the equation \(\text{f}(x) = 12\). [3]

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

1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
2. Sketch the graph of \(y = f(x)\). [3]
3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]

\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]

The line \(l\) is the tangent to \(C\) at \(O\).

1. Find an equation for \(l\). [4]
2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]