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

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

Solutions relying on calculator technology are not acceptable.

1. Using algebra, find all solutions of the equation $$3 x ^ { 3 } - 17 x ^ { 2 } - 6 x = 0$$
2. Hence find all real solutions of $$3 ( y - 2 ) ^ { 6 } - 17 ( y - 2 ) ^ { 4 } - 6 ( y - 2 ) ^ { 2 } = 0$$

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

$$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
Given

• the point \(( 2,10 )\) lies on \(C\)
• the gradient of the curve at \(( 2,10 )\) is - 3
  1. 1. show that the value of \(a\) is - 2
     2. find the value of \(b\).
  2. Hence show that \(C\) has no stationary points.
  3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
  4. Hence deduce the coordinates of the points of intersection of the curve with equation

$$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.

5. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$

1. Using the factor theorem, explain why \(\mathrm { f } ( x )\) is divisible by \(( x + 3 )\).
2. Hence fully factorise \(\mathrm { f } ( x )\).
3. Show that \(\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 3 } + 5 x ^ { 2 } + 6 x }\) can be written in the form \(A + \frac { B } { x }\) where \(A\) and \(B\) are integers to be found.

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)

2. $$f ( x ) = ( 2 x - 3 ) ( x - k ) - 12$$ where \(k\) is a constant.
a.Write down the value of \(\mathrm { f } ( k )\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) the remainder is - 5
b. find the value of \(k\).
c. Factorise \(\mathrm { f } ( x )\) completely.

7. A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 18 x ^ { 2 } + 2 a x + b\)
• the \(y\)-intercept of \(C\) is - 48
• the point \(A\), with coordinates \(( - 1,45 )\) lies on \(C\) a. Show that \(a - b = 99\) b. Find the value of \(a\) and the value of \(b\).
  c. Show that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

6. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 1\) a. (i) Show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
(ii) Express \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) ( x + a ) ^ { 2 }\) where \(a\) is an integer. Using the answer to part a) (ii)
b. show that the equation \(2 p ^ { 6 } + 3 p ^ { 4 } - 1\) has exactly two real solutions and state the values of these roots.
c. deduce the number of real solutions, for \(5 \pi \leq \theta \leq 8 \pi\), to the equation $$2 \cos ^ { 3 } \theta + 3 \cos ^ { 2 } \theta - 1 = 0$$

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.

1. Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\) The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\) Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. verify that \(b\) satisfies the equation $$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$ The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
3. Explain, with the aid of a diagram, the significance of the root 5.442

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a cubic expression in \(X\). The curve

• passes through the origin
• has a maximum turning point at \(( 2,8 )\)
• has a minimum turning point at \(( 6,0 )\)
  1. Write down the set of values of \(x\) for which

$$\mathrm { f } ^ { \prime } ( x ) < 0$$ The line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at only one point.

Find the set of values of \(k\), giving your answer in set notation.

Find the equation of \(C\). You may leave your answer in factorised form.

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 2 x ^ { 3 } + 10 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 42 x - 15 x ^ { 2 } - 7 \quad x > 0$$

1. Verify that the curves intersect at \(x = \frac { 1 } { 2 }\) The curves intersect again at the point \(P\)
2. Using algebra and showing all stages of working, find the exact \(x\) coordinate of \(P\)

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$

1. $$g ( x ) = 3 x ^ { 3 } - 20 x ^ { 2 } + ( k + 17 ) x + k$$ where \(k\) is a constant.
Given that \(( x - 3 )\) is a factor of \(\mathrm { g } ( x )\), find the value of \(k\).

1. $$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$ Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
You must make your method clear.

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant

Given that \(\mathrm { f } ( - 6 ) = 0\)

1. 1. show that \(a = 4\)
   2. express \(\mathrm { f } ( x )\) as a product of two algebraic factors. Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
2. show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
3. hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.

6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$

1. 1. Calculate f(2)
   2. Write \(\mathrm { f } ( x )\) as a product of two algebraic factors. Using the answer to (a)(ii),
2. prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
3. deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$

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

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + a x - 23\) where \(a\) is a constant
• the \(y\) intercept of \(C\) is - 12
• ( \(x + 4\) ) is a factor of \(\mathrm { f } ( x )\) find, in simplest form, \(\mathrm { f } ( x )\)

1. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + a$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).

3 In this question you must show detailed reasoning.

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 8 x + 3\).
   1. Show that \(f ( 1 ) = 0\).
   2. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Hence solve the equation \(2 \sin ^ { 3 } \theta + 3 \sin ^ { 2 } \theta - 8 \sin \theta + 3 = 0\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

3 The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}

1. State the values of \(x\) for which \(\mathrm { f } ( x ) < \frac { 1 } { 2 }\), giving your answer in set notation.
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( - x )\).
3. Explain how you can tell that \(\mathrm { f } ( x )\) cannot be expressed as the product of three real linear factors.

2 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2\), where \(a\) is a constant. You are given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Find all the factors of \(\mathrm { f } ( x )\).

1 Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).

6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).

1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).

7 In this question you must show detailed reasoning.

1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.

6 Show that the expression \(3 x ^ { 3 } + x ^ { 2 } - 6 x - 5\) can be written in the form \(( x + 2 ) \left( a x ^ { 2 } + b x + c \right) + d\) where \(a\), \(b\), \(c\) and \(d\) are constants to be determined.