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

2 Find the values of \(A , B\) and \(C\) in the identity \(2 x ^ { 2 } - 13 x + 25 \equiv A ( x - 3 ) ^ { 2 } - B ( x - 2 ) + C\).

1 Find the values of \(P , Q , R\) and \(S\) in the identity \(3 x ^ { 3 } + 18 x ^ { 2 } + P x + 31 \equiv Q ( x + R ) ^ { 3 } + S\).

2 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x ^ { 3 } - 3 \equiv ( x + 3 ) \left( A x ^ { 2 } + B x + C \right) + D\).

3 Given that \(z = 6\) is a root of the cubic equation \(z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0\), find the value of \(p\) and the other roots.

2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.

1 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x \left( x ^ { 2 } - 5 \right) \equiv ( x - 2 ) \left( A x ^ { 2 } + B x + C \right) + D\).

2 You are given that \(z = \frac { 3 } { 2 }\) is a root of the cubic equation \(2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0\). Find the other two roots.

9 The curve \(C\) with equation $$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$ where \(a , b\) and \(c\) are constants, has two asymptotes. It is given that \(y = 2 x - 5\) is one of these asymptotes.

1. State the equation of the other asymptote.
2. Find the value of \(a\) and show that \(b = - 7\).
3. Given also that \(C\) has a turning point when \(x = 2\), find the value of \(c\).
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).

4 It is given that the equation $$x ^ { 3 } - 21 x ^ { 2 } + k x - 216 = 0$$ where \(k\) is a constant, has real roots \(a , a r\) and \(a r ^ { - 1 }\).

1. Find the numerical values of the roots.
2. Deduce the value of \(k\).

State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where \(k\) is a constant, has roots \(a - d , a , a + d\). Find the numerical values of the roots and determine the value of \(k\).

10

1. Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
2. Hence explain why the roots of the equation \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) are \(x = \pm \sin \frac { 1 } { 5 } \pi\) and \(x = \pm \sin \frac { 2 } { 5 } \pi\).
3. Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).

1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.

4 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6\).

1. Use the factor theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in fully factorised form.
3. Hence solve the equation \(2 \times 8 ^ { y } - 3 \times 4 ^ { y } - 11 \times 2 ^ { y } + 6 = 0\).

7 In this question you must show detailed reasoning.
Use the substitution \(u = 6 x ^ { 2 } + x\) to solve the equation \(36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0\).

8 The number \(K\) is defined by \(K = n ^ { 3 } + 1\), where \(n\) is an integer greater than 2 .

1. Given that \(n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)\), find the constants \(b\) and \(c\).
2. Prove that \(K\) has at least two distinct factors other than 1 and \(K\).

9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(\mathrm { g } ( x ) \leqslant 0\)
   2. \(\mathrm { g } ( 2 x ) = 0\)

11. $$f ( x ) = 2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 48$$

1. Prove that \(( x - 4 )\) is a factor of \(\mathrm { f } ( x )\).
2. Hence, using algebra, show that the equation \(\mathrm { f } ( x ) = 0\) has only two distinct roots. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-24_727_1059_566_504} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
3. Deduce, giving reasons for your answer, the number of real roots of the equation $$2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 46 = 0$$ Given that \(k\) is a constant and the curve with equation \(y = \mathrm { f } ( x + k )\) passes through the origin, (d) find the two possible values of \(k\).

10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$

1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
3. Find, using algebraic integration, the exact value of the area of \(R\).

1. A curve has equation \(y = \mathrm { g } ( x )\).

Given that

• \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
• the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
• the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
  1. find \(\mathrm { g } ( x )\),
  2. prove that the stationary point at \(( 2,9 )\) is a maximum.

2. $$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Find the constants \(a\), \(b\) and \(c\) such that $$f ( x ) = ( x + 3 ) \left( a x ^ { 2 } + b x + c \right)$$
3. Hence show that \(\mathrm { f } ( x ) = 0\) has only one real root.
4. Write down the real root of the equation \(\mathrm { f } ( x - 5 ) = 0\)

1. In this question you must show detailed reasoning.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation \(y = 8 - 10 x + 6 x ^ { 2 } - x ^ { 3 }\) The curve \(C _ { 2 }\) has equation \(y = x ^ { 2 } - 12 x + 14\)

1. Verify that when \(x = 1\) the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect. The curves also intersect when \(x = k\).
   Given that \(k < 0\)
2. use algebra to find the exact value of \(k\).

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where \(a\) and \(b\) are constants.
Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
2. express \(\mathrm { f } ( x )\) in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where \(k\) is a constant to be found.
3. Hence,
   1. state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
   2. write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)

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 $$y = 3 x ^ { 2 } - 4 x - 10$$
   (b) 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 \(- \frac { 2 } { 3 }\)
2. 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.)