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.

8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.

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 $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).

Find the quotient and remainder when \(x^4\) is divided by \(x^2 + 2x - 1\). [3]

4 The polynomial \(x ^ { 3 } + 3 x ^ { 2 } + 4 x + 2\) is denoted by \(\mathrm { f } ( x )\).

1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + x - 1\).
2. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

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 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.

Given that \(f(x) = (x^2 - 6x)(x - 2) + 3x\),

1. express \(f(x)\) in the form \(a(x^2 + bx + c)\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Hence factorise \(f(x)\) completely. [2]
3. Sketch the graph of \(y = f(x)\), showing the coordinates of each point at which the graph meets the axes. [3]

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 .

1. Express \(\frac{3 + 13x - 6x^2}{2x - 3}\) in the form \(Ax + B + \frac{C}{2x - 3}\). [4 marks]
2. Show that \(\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3\), where \(p\) and \(q\) are rational numbers. [4 marks]

7

1. Find the quotient when \(9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1\) is divided by ( \(3 x + 2\) ), and show that the remainder is 9 .
2. Hence find \(\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x\), giving the answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
3. Find the exact root of the equation \(9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Find the quotient when \(9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1\) is divided by ( \(3 x + 2\) ), and show that the remainder is 9 .
2. Hence find \(\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x\), giving the answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
3. Find the exact root of the equation \(9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 A curve has equation \(y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }\).

1. Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.
2. State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
3. Sketch the curve.
4. Solve the inequality \(\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0\).

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 Fig. 7 shows an incomplete sketch of \(y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }\) where \(a , b\) and \(c\) are integers. The asymptotes of the curve are also shown. \begin{figure}[h] \end{figure}

1. Determine the values of \(a , b\) and \(c\). Use these values of \(a , b\) and \(c\) throughout the rest of the question.
2. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\), justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.
3. Find the \(x\) coordinates of the points on the curve where \(y = 1\). Write down the solution to the inequality \(\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1\).

The expression $$\frac{12x^2 + 3x + 7}{3x - 5}$$ can be written as $$Ax + B + \frac{C}{3x - 5}$$ State the value of \(A\) Circle your answer. [1 mark] $$3 \quad 4 \quad 7 \quad 9$$

Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2x^3 - 3x^2 + x - 2 \equiv (x + 2)(Ax^2 + Bx + C) + D.$$ [5]

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

3 The polynomial \(x ^ { 4 } - 4 x ^ { 3 } + 3 x ^ { 2 } + 4 x - 4\) is denoted by \(\mathrm { p } ( x )\).

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - 3 x + 2\).
2. Hence solve the equation \(\mathrm { p } ( x ) = 0\).

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 \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)

3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\) the remainder is \(2 x + 3\). Find the values of \(a\) and \(b\).

4

1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.

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 \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).

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

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.

Express as a single fraction in its simplest form $$\frac{x^2 - 8x + 15}{x^2 - 9} \times \frac{2x^2 + 6x}{(x - 5)^2}.$$ [4]

7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.

4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).

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.

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.

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.

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 $$p ( x ) = 9 x ^ { 3 } + 6 x ^ { 2 } + 12 x + k$$ where \(k\) is a constant.

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\) and show that the remainder is \(( k - 8 )\).
2. It is given that \(\int _ { 1 } ^ { 6 } \frac { \mathrm { p } ( \mathrm { x } ) } { 3 \mathrm { x } + 2 } \mathrm { dx } = \mathrm { a } + \ln 64\), where \(a\) is an integer. Find the values of \(a\) and \(k\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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

7 A curve has equation \(y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }\).

1. Write down the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three asymptotes.
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.

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

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.