Polynomial intersection with algebra

11. (a) On Diagram 1 sketch the graphs of

1. \(y = x ( 3 - x )\)
2. \(y = x ( x - 2 ) ( 5 - x )\)
   showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\) coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
   (c) find, using algebra and showing your working, the exact coordinates of \(P\).
   
   \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
   \section*{Diagram 1}

13. (a) On separate axes sketch the graphs of

1. \(y = c ^ { 2 } - x ^ { 2 }\)
2. \(y = x ^ { 2 } ( x - 3 c )\)
   where \(c\) is a positive constant.
   Show clearly the coordinates of the points where each graph crosses or meets the \(x\)-axis and the \(y\)-axis.
   (b) Prove that the \(x\) coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where \(c\) is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when \(x = 2\)
   (c) find the exact value of \(c\), writing your answer as a fully simplified surd.

10. (a) On the same axes sketch the graphs of the curves with equations

1. \(y = x ^ { 2 } ( x - 2 )\),
2. \(y = x ( 6 - x )\),
   and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
   (b) Use algebra to find the coordinates of the points where the graphs intersect.

10. (a) On the axes below sketch the graphs of

1. \(y = x ( 4 - x )\)
2. \(y = x ^ { 2 } ( 7 - x )\)
   showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
   (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers.
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2 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

4

1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).

1. (a) On the same diagram, sketch

$$y = ( x + 1 ) ( 2 - x ) \quad \text { and } \quad y = x ^ { 2 } - 2 | x |$$ Mark clearly the coordinates of the points where these curves cross the coordinate axes.
(b) Find the \(x\)-coordinates of the points of intersection of these two curves.

1. (a) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).

Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
(b) Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$ (c) State the integer, \(n\), such that $$n < \alpha < n + 1 .$$