Algebraic Fraction Simplification

8. (a) Simplify fully $$\frac { 2 x ^ { 2 } + 9 x - 5 } { x ^ { 2 } + 2 x - 15 }$$ Given that $$\ln \left( 2 x ^ { 2 } + 9 x - 5 \right) = 1 + \ln \left( x ^ { 2 } + 2 x - 15 \right) , \quad x \neq - 5$$ (b) find \(x\) in terms of e.

1. Express

$$\frac { 3 x ^ { 2 } } { \left( 2 x ^ { 2 } + 7 x + 6 \right) } \times \frac { 7 ( 3 + 2 x ) } { 3 x ^ { 5 } }$$ as a single fraction in its simplest form.

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

10 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$

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 Simplify \(\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }\).

1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express \(\mathrm { f } ( x )\) in its simplest form.

1. Express

$$\frac { 2 x } { 2 x ^ { 2 } + 3 x - 5 } \div \frac { x ^ { 3 } } { x ^ { 2 } - x }$$ as a single fraction in its simplest form.

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.

4 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$

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

1 Simplify \(\frac { x ^ { 4 } - 10 x ^ { 2 } + 9 } { \left( x ^ { 2 } - 2 x - 3 \right) \left( x ^ { 2 } + 8 x + 15 \right) }\).

1. Express as a single fraction in its simplest form

$$\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 } \times \frac { 2 x ^ { 2 } + 6 x } { ( x - 5 ) ^ { 2 } }$$

1. (a) Simplify \(\frac { x ^ { 2 } + 4 x + 3 } { x ^ { 2 } + x }\).
   (b) Find the value of \(x\) for which \(\log _ { 2 } \left( x ^ { 2 } + 4 x + 3 \right) - \log _ { 2 } \left( x ^ { 2 } + x \right) = 4\).
2. (i) Prove, by counter-example, that the statement

$$\text { " } \sec ( A + B ) \equiv \sec A + \sec B , \text { for all } A \text { and } B \text { " }$$ is false
(ii) Prove that $$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$

2

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8\).
   1. Use the Factor Theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
   2. Write \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 2 } + 16 x } { 2 x ^ { 3 } + 3 x ^ { 2 } - 18 x + 8 }\).
2. Express the algebraic fraction \(\frac { 2 x ^ { 2 } } { ( x + 5 ) ( x - 3 ) }\) in the form \(A + \frac { B + C x } { ( x + 5 ) ( x - 3 ) }\), where \(A , B\) and \(C\) are integers.

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
   1. Find \(\mathrm { f } ( - 1 )\).
   2. Use the Factor Theorem to show that \(2 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 4 x ^ { 3 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(2 x + 1\), the remainder is 2 . Find the value of \(d\).

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 15 x ^ { 3 } + 19 x ^ { 2 } - 4\).

1. 1. Find \(\mathrm { f } ( - 1 )\).
   2. Show that \(( 5 x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.

1

1. The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10\).
   1. Find \(\mathrm { p } ( 2 )\).
   2. Use the Factor Theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
   3. Write \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Hence simplify \(\frac { 3 x ^ { 2 } - 6 x } { 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10 }\).

1

1. Find the remainder when \(2 x ^ { 2 } + x - 3\) is divided by \(2 x + 1\).
   (2 marks)
2. Simplify the algebraic fraction \(\frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - 1 }\).
   (3 marks)

2 Simplify fully \(\frac { 2 x ^ { 3 } + x ^ { 2 } - 7 x - 6 } { x ^ { 2 } - x - 2 }\).

4

1. Use the factor theorem to prove that \(x + 3\) is a factor of \(\mathrm { p } ( x )\) [0pt] [2 marks] 4
2. Simplify the expression \(\frac { 2 x ^ { 3 } + 7 x ^ { 2 } + 2 x - 3 } { 4 x ^ { 2 } - 1 } , x \neq \pm \frac { 1 } { 2 }\) [0pt] [4 marks]