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

5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$

1. In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = x ^ { 2 } ( 2 x + 1 ) - 15 x$$

1. Solve $$\mathrm { f } ( x ) = 0$$
2. Hence solve $$y ^ { \frac { 4 } { 3 } } \left( 2 y ^ { \frac { 2 } { 3 } } + 1 \right) - 15 y ^ { \frac { 2 } { 3 } } = 0 \quad y > 0$$ giving your answer in simplified surd form.

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

\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.

1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
2. Hence, using algebra, find the range of possible values of \(p\)

6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) there is no remainder. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 12

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find $$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$ simplifying your answer.

8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where \(p\) and \(q\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is - 6

1. Use the remainder theorem to show that \(p + q = - 5\) Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(p\) and the value of \(q\).
3. Factorise \(\mathrm { f } ( \mathrm { x } )\) completely.
   

5. $$f ( x ) = x ^ { 3 } + ( p + 3 ) x ^ { 2 } - x + q$$ where \(p\) and \(q\) are constants and \(p > 0\) Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\)

1. show that $$9 p + q = - 51$$ Given also that when \(\mathrm { f } ( x )\) is divided by ( \(x + p\) ) the remainder is 9
2. show that $$3 p ^ { 2 } + p + q - 9 = 0$$
3. Hence find the value of \(p\) and the value of \(q\).
4. Hence find a quadratic expression \(\mathrm { g } ( x )\) such that $$f ( x ) = ( x - 3 ) g ( x )$$

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

Solutions relying on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + a x ^ { 2 } - 29 x + b$$ where \(a\) and \(b\) are constants.
Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a + 4 b = - 56$$ Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 25
2. find a second simplified equation linking \(a\) and \(b\).
3. Hence, using algebra and showing your working,
   1. find the value of \(a\) and the value of \(b\),
   2. fully factorise \(\mathrm { f } ( x )\).

1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ), the remainder is 7

1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
2. Find the value of \(a\) and the value of \(b\)

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3. $$f ( x ) = ( 3 x - 2 ) ( x - k ) - 8$$ where \(k\) is a constant.

1. Write down the value of \(\mathrm { f } ( k )\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 4
2. Find the value of \(k\).
3. Factorise f(x) completely.

4. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 10 x + 24$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise f(x) completely.

4. \(\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ) the remainder is 55
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 9

1. Find the value of \(a\) and the value of \(b\). Given that \(( 3 x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. factorise \(\mathrm { f } ( x )\) completely.

3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that \(a = - 9\)
2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.

3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,

1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(A\) and the value of \(B\).
3. Factorise f(x) fully.

4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
3. Factorise \(\mathrm { f } ( x )\) completely.

6. $$f ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 40 x + 21$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\)
2. Factorise f(x) completely.
3. Hence solve the equation $$6 \left( 2 ^ { 3 y } \right) + 7 \left( 2 ^ { 2 y } \right) = 40 \left( 2 ^ { y } \right) + 21$$ giving your answer to 2 decimal places.

3. $$f ( x ) = 3 - \frac { x - 2 } { x + 1 } + \frac { 5 x + 26 } { 2 x ^ { 2 } - 3 x - 5 } \quad x > 4$$

1. Show that $$\mathrm { f } ( x ) = \frac { a x + b } { c x + d } \quad x > 4$$ where \(a , b , c\) and \(d\) are integers to be found.
2. Hence find \(\mathrm { f } ^ { - 1 } ( x )\)
3. Find the domain of \(\mathrm { f } ^ { - 1 }\)

6. The function f is defined by $$f ( x ) = \frac { 5 x - 3 } { x - 4 } \quad x > 4$$

1. Show, by using calculus, that f is a decreasing function.
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Show that \(\mathrm { ff } ( x ) = \frac { a x + b } { x + c }\) where \(a , b\) and \(c\) are constants to be found.
   2. Deduce the range of ff.

1. The functions f and g are defined by

$$\begin{array} { l l l } \mathrm { f } ( x ) = 9 - x ^ { 2 } & x \in \mathbb { R } & x \geqslant 0 \\ \mathrm {~g} ( x ) = \frac { 3 } { 2 x + 1 } & x \in \mathbb { R } & x \geqslant 0 \end{array}$$

1. Write down the range of f
2. Find the value of fg(1.5)
3. Find \(\mathrm { g } ^ { - 1 }\)

1. The function f is defined by

$$f ( x ) = \frac { 2 x ^ { 2 } - 32 } { 3 x ^ { 2 } + 7 x - 20 } + \frac { 8 } { 3 x - 5 } \quad x \in \mathbb { R } \quad x > 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 5 }\)
2. Show, using calculus, that f is a decreasing function. You must make your reasoning clear. The function g is defined by $$g ( x ) = 3 + 2 \ln x \quad x \geqslant 1$$
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 5$$

1. (i) Find

$$\int \frac { 12 } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) (a) Write \(\frac { 4 x + 3 } { x + 2 }\) in the form $$A + \frac { B } { x + 2 } \text { where } A \text { and } B \text { are constants to be found }$$ (b) Hence find, using algebraic integration, the exact value of $$\int _ { - 8 } ^ { - 5 } \frac { 4 x + 3 } { x + 2 } d x$$ giving your answer in simplest form.

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }\)
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.
      (3)

1. Express

$$\frac { 6 x + 4 } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$ as a single fraction in its simplest form.

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { 3 x - 4 } { x - 3 } , \quad x \in \mathbb { R } , \quad x < 3$$ The graph cuts the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 2 .

1. State the range of g .
2. State the coordinates of
   1. point \(A\)
   2. point \(B\)
3. Find \(\operatorname { gg } ( x )\) in its simplest form.
4. Sketch the graph with equation \(y = | \mathrm { g } ( x ) |\) On your sketch, show the coordinates of each point at which the graph meets or cuts the axes and state the equation of each asymptote.
5. Find the exact solution of the equation \(| \mathrm { g } ( x ) | = 8\)

8. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-26_679_1168_733_390} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
3. Calculate the range of \(\mathrm { h } ( x )\).