Simple 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.

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

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

5

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

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.

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

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

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.

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)

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

It is given that $$f(x) = \frac{x^2 + 2x - 24}{x^2 - 4x} \quad \text{for } x \neq 0, x \neq 4.$$ Express \(f(x)\) in its simplest form. [3]

Simplify fully \(\frac{2x^3 + x^2 - 7x - 6}{x^2 - x - 2}\). [4]