1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

14. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).

1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.

9. \(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant (a) Write down the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + k )\). When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 15
(b) Show that \(k = 2\) (c) Hence factorise \(\mathrm { f } ( x )\) completely. \section*{9.} " . \(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant

4. $$f ( x ) = 6 x ^ { 3 } - 7 x ^ { 2 } - 43 x + 30$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
   1. \(2 x + 1\)
   2. \(x - 3\)
2. Hence factorise \(\mathrm { f } ( x )\) completely.

12. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).

1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
3. Write down the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}

7. $$g ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 18 x - 8$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\),

1. show that \(a = - 3\)
2. Hence, using algebra, fully factorise \(\mathrm { g } ( x )\). Using your answer to part (b),
3. solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$2 \sin ^ { 3 } \theta - 3 \sin ^ { 2 } \theta - 18 \sin \theta = 8$$ giving each answer, in radians, as a multiple of \(\pi\).
   

1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
(5)
VILIV SIMI NI III IM I ON OC
VILV SIHI NI JAHMMION OC
VALV SIHI NI JIIIM ION OC

8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where \(p\) and \(q\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is - 6

1. Use the remainder theorem to show that \(p + q = - 5\) Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(p\) and the value of \(q\).
3. Factorise \(\mathrm { f } ( \mathrm { x } )\) completely.
   

6. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7

1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
2. Find the value of \(a\) and the value of \(b\).

1. Factorise completely

$$x ^ { 3 } - 4 x ^ { 2 } + 3 x .$$

9. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.

1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
4. Find the exact coordinates of \(R\).

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.

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.

8. The point \(P ( 1 , a )\) lies on the curve with equation \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\).

1. Find the value of \(a\).
2. On the axes below sketch the curves with the following equations:
   1. \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\),
   2. \(y = \frac { 2 } { x }\). On your diagram show clearly the coordinates of any points at which the curves meet the axes.
3. With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
   \includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}

Factorise completely \(x - 4 x ^ { 3 }\)

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point ( 3,6 ). Given that $$f ^ { \prime } ( x ) = ( x - 2 ) ( 3 x + 4 )$$

1. use integration to find \(\mathrm { f } ( x )\). Give your answer as a polynomial in its simplest form.
2. Show that \(\mathrm { f } ( x ) \equiv ( x - 2 ) ^ { 2 } ( x + p )\), where \(p\) is a positive constant. State the value of \(p\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the curve touches or crosses the coordinate axes.

9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),

1. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
2. Hence factorise \(\mathrm { f } ( x )\) completely.
3. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.

9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),

1. use integration to find \(\mathrm { f } ( x )\).
2. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
3. In the space provided on page 17, sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}

10.

1. Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
2. Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
3. sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.
   

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. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).

1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
   Calculate the values of \(a , b\) and \(c\).
2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(( 1,0 )\), touches it at \(( - 3,0 )\) and crosses the \(y\)-axis at \(( 0 , - 9 )\)

1. In the space below, sketch the curve \(C\) with equation \(y = \mathrm { f } ( x + 2 )\) and state the coordinates of the points where the curve \(C\) meets the \(x\)-axis.
2. Write down an equation of the curve \(C\).
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C\) meets the \(y\)-axis.

9. $$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }\),
   where \(A\) and \(B\) are constants to be found.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\). Given that the point \(( - 3,10 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\),
3. find \(\mathrm { f } ( x )\).

Factorise fully \(25 x - 9 x ^ { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-02_37_42_2700_1909}