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

7

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 7 \theta = \frac { \operatorname { cosec } ^ { 7 } \theta } { 7 \operatorname { cosec } ^ { 6 } \theta - 56 \operatorname { cosec } ^ { 4 } \theta + 112 \operatorname { cosec } ^ { 2 } \theta - 64 }$$
2. Hence obtain the roots of the equation $$x ^ { 7 } - 14 x ^ { 6 } + 112 x ^ { 4 } - 224 x ^ { 2 } + 128 = 0$$ in the form \(\operatorname { cosec } q \pi\), where \(q\) is rational.

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
2. Expand f(x) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found. The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis. The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
3. Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
   (Solutions relying entirely on calculator technology are not acceptable.)

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 4 ) ( x - 2 ) ( 2 x - 9 )$$ Given that the curve with equation \(y = \mathrm { f } ( x ) - p\) passes through the point with coordinates \(( 0,50 )\)

1. find the value of the constant \(p\). Given that the curve with equation \(y = \mathrm { f } ( x + q )\) passes through the origin,
2. write down the possible values of the constant \(q\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the range of values of \(x\) for which the gradient of the curve with equation \(y = \mathrm { f } ( x )\) is less than - 18 \includegraphics[max width=\textwidth, alt={}, center]{6c320b71-8793-461a-a078-e4f64c144a3a-23_68_37_2617_1914}

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$

1. Use the given information to state the values of \(x\) for which $$f ( x ) > 0$$
2. Expand \(( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )\), writing your answer as a polynomial in simplest form. The straight line \(l\) is the tangent to \(C\) at the point where \(C\) cuts the \(y\)-axis.
   Given that \(l\) cuts \(C\) at the point \(P\), as shown in Figure 4,
3. find, using algebra, the \(x\) coordinate of \(P\) (Solutions based on calculator technology are not acceptable.)

1. Find

$$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\) Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\) Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)

2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = a x ^ { 3 } + ( 6 a + 8 ) x ^ { 2 } - a ^ { 2 } x$$ where \(a\) is a positive constant. Given \(\mathrm { f } ( - 1 ) = 32\)

1. 1. show that the only possible value for \(a\) is 3
   2. Using \(a = 3\) solve the equation $$\mathrm { f } ( x ) = 0$$
2. Hence find all real solutions of
   1. \(3 y + 26 y ^ { \frac { 2 } { 3 } } - 9 y ^ { \frac { 1 } { 3 } } = 0\)
   2. \(3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0\)

8. The curve \(C _ { 1 }\) has equation $$y = 3 x ^ { 2 } + 6 x + 9$$

1. Write \(3 x ^ { 2 } + 6 x + 9\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The point \(P\) is the minimum point of \(C _ { 1 }\)
2. Deduce the coordinates of \(P\). A different curve \(C _ { 2 }\) has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where \(A\), \(B\), \(C\) and \(D\) are constants. Given that \(C _ { 2 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
4. find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}

10. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
3. Hence, or otherwise, find
   1. the coordinates of \(P\),
   2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
4. 1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
   2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).

1. The curve \(C\) has equation

$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(R \left( 2 , \frac { 13 } { 2 } \right)\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(R\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 1 } { x + 2 }\)

1. State the equation of the asymptote of \(C\) that is parallel to the \(y\)-axis.
2. Factorise fully \(x ^ { 3 } + 4 x ^ { 2 } + 4 x\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
3. On Diagram 1, add a sketch of the curve with equation $$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$ On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
4. Hence state the number of real solutions of the equation $$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$ giving a reason for your answer.
   \includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
   Only use the copy of Diagram 1 if you need to redraw your answer to part (c).

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve \(C _ { 2 }\) has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$

1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at points \(P\) and \(Q\)
2. find, using algebra, the exact distance \(P Q\)

3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$

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

10. $$f ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 5$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) there is no remainder.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is - 15

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

6. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\).
3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)

8. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } - 23 x - 10$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ).
2. Show that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\).
4. Hence solve $$2 \left( 3 ^ { 3 t } \right) - 5 \left( 3 ^ { 2 t } \right) - 23 \left( 3 ^ { t } \right) = 10$$ giving your answer to 3 decimal places.

6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) there is no remainder. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 12

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

13. \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12\), where \(c\) is a constant Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that \(c = - 14\)
2. Write \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where \(\mathrm { Q } ( x )\) is a quadratic function.
3. Use the answer to part (b) to prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real solution. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). On separate diagrams sketch the curve with equation
4. 1. \(y = \mathrm { f } ( 3 x )\)
   2. \(y = - \mathrm { f } ( \mathrm { x } )\) On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.

2. $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).

7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that \(2 a + b = - 7\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
2. find the value of \(a\) and the value of \(b\). \(\mathrm { f } ( x )\) can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
3. 1. Find \(\mathrm { Q } ( x )\).
   2. Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. You must justify your answer and show all your working.

5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ).
2. Use the factor theorem to show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
4. Use your answer to part (c) and the sketch to deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)

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

1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
2. Find the value of \(a\) and the value of \(b\).