Polynomial Division & Manipulation

1 Simplify \(\frac { 20 - 5 x } { 6 x ^ { 2 } - 24 x }\).

9

1. Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
2. Factorise \(x ^ { 2 } - 4\) and \(x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.

3 Factorise and hence simplify \(\frac { 3 x ^ { 2 } - 7 x + 4 } { x ^ { 2 } - 1 }\).

1. Prove that 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all even positive integers \(n\).
2. Determine whether 12 is a factor of \(3 n ^ { 2 } + 6 n\) for all positive integers \(n\).
3. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
4. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
5. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
6. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

1 Find the quotient and remainder when \(2 x ^ { 2 }\) is divided by \(x + 2\).

1 Find the quotient and remainder when \(2 x ^ { 2 }\) is divided by \(x + 2\).

2 The polynomial \(x ^ { 3 } + 2 x ^ { 2 } + 2 x + 3\) is denoted by \(\mathrm { p } ( x )\).

1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\).

2 A curve has equation \(y = \frac { x ^ { 2 } - 6 x - 5 } { x - 2 }\).

1. Find the equations of the asymptotes.
2. Show that \(y\) can take all real values.

7 A curve has equation \(y = \frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } }\).

1. Show that \(y\) can never be zero.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
4. Sketch the curve.
5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).

7. (a) Find the set of values of \(k\) for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }\) The curve has asymptotes \(x = a , x = b\) and \(y = c\), where \(a , b\) and \(c\) are integers.
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\).
(c) Find the coordinates of the points of intersection of \(C _ { 1 }\) with the line \(y = 2\) (d) Find all the integer pairs \(( r , s )\) that satisfy \(s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }\) The curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\) where \(\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }\) (e) Show that, for suitable integers \(m\) and \(n , \mathrm {~g} ( x )\) can be written in the form \(\mathrm { f } ( x + m ) + n\).
(f) Sketch \(C _ { 2 }\) showing any asymptotes and stating their equations.

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where \(k\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points of \(C\).
3. Sketch \(C\).

4. $$\mathrm { g } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } , \quad x > 3 , \quad x \in \mathbb { R }$$

1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } \equiv x ^ { 2 } + A + \frac { B } { x - 3 }$$ find the values of the constants \(A\) and \(B\).
2. Hence, or otherwise, find the equation of the tangent to the curve with equation \(y = \mathrm { g } ( x )\) at the point where \(x = 4\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
   (5)
   

The curve \(C\) has equation $$y = \frac { x ( x + 1 ) } { ( x - 1 ) ^ { 2 } }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that there is exactly one point of intersection of \(C\) with the asymptotes and find its coordinates.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence
   (a) find the coordinates of any stationary points of \(C\),
   (b) state the set of values of \(x\) for which the gradient of \(C\) is negative.
4. Draw a sketch of \(C\).

4 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 7 x + 6 } { x - 2 }$$

1. Find the coordinates of the points of intersection of \(C\) with the axes.
2. Find the equation of each of the asymptotes of \(C\).
3. Sketch C.

8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( x - 3 ) ( x + 1 ) ( x - 1 ) }\).

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
3. Determine whether the curve approaches the horizontal asymptote from above or below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
   Fully justify your answer.
   9
2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
   \section*{Question 9 continues on the next page} 9
4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
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2 Two curves have equations \(y = \ln x\) and \(y = \frac { k } { x }\), where \(k\) is a positive constant.

1. Sketch the curves on a single diagram.
2. Explain how your diagram shows that the equation \(x \ln x - k = 0\) has exactly one real root.

13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).

12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Find the other factors of \(\mathrm { f } ( x )\).
3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
   Solve this equation to find the \(x\)-coordinates of A and B .

3 When \(x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a\) is divided by \(x ^ { 2 } + 2 x - 1\), the quotient is \(x ^ { 2 } + b x + 2\) and the remainder is \(c x + 7\). Find the values of the constants \(a , b\) and \(c\).

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$ where \(a\) is a constant.

1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + a x + 2 a$$ where \(a\) is an integer.

1. Find, in terms of \(x\) and \(a\), the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 4 .
2. It is given that \(\int _ { - 1 } ^ { 1 } \frac { \mathrm { p } ( x ) } { x + 2 } \mathrm {~d} x = \frac { 22 } { 3 } + \ln b\), where \(b\) is an integer. Find the values of \(a\) and \(b\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 Expand and simplify \(( n + 2 ) ^ { 3 } - n ^ { 3 }\).

3 Simplify \(\frac { \left( 3 x y ^ { 4 } \right) ^ { 3 } } { 6 x ^ { 5 } y ^ { 2 } }\).

5

1. Expand and simplify \(( 2 x + 1 ) ( x - 3 ) ( x + 4 )\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of $$x \left( x ^ { 2 } + 2 x + 3 \right) \left( x ^ { 2 } + 7 x - 2 \right) .$$

5

1. Simplify \(\frac { x ^ { 3 } - x ^ { 2 } - 3 x - 9 } { x - 3 }\).
2. Hence or otherwise solve the equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 9 = 6 ( x - 3 )\).

2

1. Find the quotient when \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x\) is divided by \(x ^ { 2 } + 5 x + 6\), and show that the remainder is 18 .
2. Hence solve the equation \(4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0\).

3

1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).

8 A curve has equation \(y = \frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) }\).

1. Write down the equations of the three asymptotes.
2. Determine whether the curve approaches the horizontal asymptote from above or below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
3. Sketch the curve.
4. Solve the inequality \(\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0\).

3 Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x ^ { 3 } - 4 \equiv ( x - 1 ) \left( A x ^ { 2 } + B x + C \right) + D .$$

9

1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).

3 The expression $$\frac { 12 x ^ { 2 } + 3 x + 7 } { 3 x - 5 }$$ can be written as $$A x + B + \frac { C } { 3 x - 5 }$$ State the value of \(A\) Circle your answer.
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2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$

1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
2. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)

1. Express \(\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }\) as a single fraction in its simplest form.
   

Factorise completely $$x ^ { 3 } - 9 x .$$

5

1. Find the quotient when \(x ^ { 4 } - 32 x + 55\) is divided by \(( x - 2 ) ^ { 2 }\) and show that the remainder is 7 .
2. Factorise \(x ^ { 4 } - 32 x + 48\).
3. Hence solve the equation \(\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0\), giving your answer in an exact form.

3

1. Find the quotient when the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 13$$ is divided by \(4 x ^ { 2 } + 4 x - 3\), and show that the remainder is 4 .
2. Hence, or otherwise, factorise the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 9$$

1 When the polynomial \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + 1\), the quotient is \(x ^ { 2 } + 4 x + 2\) and the remainder is \(x - 1\). Find \(\mathrm { f } ( x )\), simplifying your answer.