Division then Solve Polynomial Equation

3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).

1. The curve \(C _ { 1 }\) has equation

$$y = \frac { 6 } { x } + 3$$

1. 1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
   2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
2. Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
3. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying on calculator technology are not acceptable.)

13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).

1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.

4. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
2. find the coordinates of \(Q\) and \(R\).

\includegraphics{figure_2} Figure 2 shows the curves with equations \(y = 7 - 2x - 3x^2\) and \(y = \frac{2}{x}\). The two curves intersect at the points \(P\), \(Q\) and \(R\).

1. Show that the \(x\)-coordinates of \(P\), \(Q\) and \(R\) satisfy the equation $$3x^3 + 2x^2 - 7x + 2 = 0.$$ [2] Given that \(P\) has coordinates \((-2, -1)\),
2. find the coordinates of \(Q\) and \(R\). [6]

In this question you must show detailed reasoning. The curve \(C_1\) has equation \(y = 8 - 10x + 6x^2 - x^3\) The curve \(C_2\) has equation \(y = x^2 - 12x + 14\)

1. Verify that when \(x = 1\) the curves \(C_1\) and \(C_2\) intersect. [2]

The curves also intersect when \(x = k\). Given that \(k < 0\)

1. use algebra to find the exact value of \(k\). [5]