1.02k Simplify rational expressions: factorising, cancelling, algebraic division

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.

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.

5. a. Given that $$\frac { x ^ { 2 } - 1 } { x + 3 } \equiv x + P + \frac { Q } { x + 3 }$$ find the value of the constant \(P\) and show that \(Q = 8\) \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 } \quad x > - 3$$ Figure 3 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 5\).
b. Find the exact area of \(R\), writing your answer in the form \(a \ln 2\), where \(a\) is constant to be found.
(4)

1. The figure 1 shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { a x + 4 } { x - b } , \quad x > 2\)

\begin{figure}[h] \end{figure} a. State the values of \(a\) and \(b\).
b. State the range of f.
c. Find \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.

7

1. Express \(\frac { 1 } { x } + \frac { 1 } { A - x }\) as a single fraction. The population of fish in a lake is modelled by the differential equation \(\frac { d x } { d t } = \frac { x ( 400 - x ) } { 400 }\) where \(x\) is the number of fish and \(t\) is the time in years.
   When \(t = 0 , x = 100\).
2. In this question you must show detailed reasoning. Find the number of fish in the lake when \(t = 10\), as predicted by the model.

1 Express \(\frac { 2 } { x - 1 } + \frac { 5 } { 2 x + 1 }\) as a single fraction.

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

1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END

7 A curve is defined, for \(x > 0\), by the equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$

1. Express \(\frac { x ^ { 8 } - 1 } { x ^ { 3 } }\) in the form \(x ^ { p } - x ^ { q }\), where \(p\) and \(q\) are integers.
2. 1. Hence differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence show that f is an increasing function.
3. Find the gradient of the normal to the curve at the point \(( 1,0 )\).

9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).

1. Find the value of \(k\).
2. Find the value of \(r\).
3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\). END

\(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 } { x ^ { 2 } + x - 6 }\).

1. Using algebraic division, show that $$f ( x ) = x ^ { 2 } + A + \frac { B } { x + C }$$ where \(A , B\) and \(C\) are integers to be found.
2. By sketching two suitable graphs on the same set of axes, show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root.
3. Use the iterative formula $$x _ { n + 1 } = 2 + \frac { 1 } { x _ { n } ^ { 2 } + 1 } ,$$ with a suitable starting value to find the root of the equation \(\mathrm { f } ( x ) = 0\) correct to 3 significant figures and justify the accuracy of your answer.

5. $$f ( x ) = 3 - \frac { x - 1 } { x - 3 } + \frac { x + 11 } { 2 x ^ { 2 } - 5 x - 3 } , \quad x \in \mathbb { R } , \quad x < - 1$$

1. Show that $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 } , \quad x \in \mathbb { R } , \quad x < 1$$

1. Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
2. Find an expression for the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
3. Solve the equation \(\mathrm { f } ( x ) = 2\).

5.

1. Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
2. Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
3. Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$

3.

1. Simplify $$\frac { 2 x ^ { 2 } + 3 x - 9 } { 2 x ^ { 2 } - 7 x + 6 }$$
2. Solve the equation $$\ln \left( 2 x ^ { 2 } + 3 x - 9 \right) = 2 + \ln \left( 2 x ^ { 2 } - 7 x + 6 \right)$$ giving your answer in terms of e.
   

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
   1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).
   (2 marks)
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + d\), where \(d\) is a constant.
   1. Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { g } ( x )\), show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4\).
      (1 mark)
   2. Given that \(\mathrm { g } ( x )\) can be written as \(\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)\), where \(a\) is an integer, express \(\mathrm { g } ( x )\) as a product of three linear factors.
   3. Hence, or otherwise, show that \(\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }\), where \(p\) and \(q\) are integers.
      (3 marks)

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 13 x + 6\).

1. Find \(\mathrm { f } ( - 2 )\).
2. Use the Factor Theorem to show that \(2 x - 3\) is a factor of \(\mathrm { f } ( x )\).
3. Simplify \(\frac { 2 x ^ { 2 } + x - 6 } { \mathrm { f } ( x ) }\).

1

1. 1. Express \(\frac { 5 x - 6 } { x ( x - 3 ) }\) in the form \(\frac { A } { x } + \frac { B } { x - 3 }\).
      (2 marks)
   2. Find \(\int \frac { 5 x - 6 } { x ( x - 3 ) } \mathrm { d } x\).
      (2 marks)
2. 1. Given that $$4 x ^ { 3 } + 5 x - 2 = ( 2 x + 1 ) \left( 2 x ^ { 2 } + p x + q \right) + r$$ find the values of the constants \(p , q\) and \(r\).
   2. Find \(\int \frac { 4 x ^ { 3 } + 5 x - 2 } { 2 x + 1 } \mathrm {~d} x\).

2

1. Given that \(\frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }\) can be expressed as \(A x + \frac { B ( 4 x - 1 ) } { 2 x ^ { 2 } - x + 2 }\), find the values of the constants \(A\) and \(B\).
2. The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$$ The point \(( - 1,2 )\) lies on the curve. Find the equation of the curve.
   [0pt] [4 marks]

3

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d\), where \(d\) is a constant. When \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ), the remainder is - 2 . Use the Remainder Theorem to find the value of \(d\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3\).
   1. Given that \(x = - \frac { 1 } { 2 }\) is a solution of the equation \(\mathrm { g } ( x ) = 0\), write \(\mathrm { g } ( x )\) as a product of three linear factors.
   2. The function h is defined by \(\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }\) for \(x > 2\). Simplify \(\mathrm { h } ( x )\), and hence show that h is a decreasing function.
      [0pt] [4 marks]

5 A curve has equation $$y = \frac { x } { x ^ { 2 } - 1 }$$

1. Write down the equations of the three asymptotes to the curve.
2. Sketch the curve.
   (You are given that the curve has no stationary points.)
3. Solve the inequality $$\frac { x } { x ^ { 2 } - 1 } > 0$$

6 A curve has equation $$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the equations of the three asymptotes of this curve.
   2. State the coordinates of the points at which the curve intersects the \(x\)-axis.
   3. Sketch the curve.
      (You are given that the curve has no stationary points.)
2. Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$