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

333 questions

Sort by: Default | Easiest first | Hardest first
Edexcel C3 Q25
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2 (x^2 + 4x + 3) - \log_2 (x^2 + x) = 4\). [4]
Edexcel C3 Q29
6 marks Moderate -0.3
  1. Express as a fraction in its simplest form $$\frac{2}{x-3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
  2. Hence solve $$\frac{2}{x-3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]
Edexcel C3 Q33
5 marks Moderate -0.3
The function \(f\) is given by \(f: x \mapsto 2 + \frac{3}{x + 2}, x \in \mathbb{R}, x \neq -2\).
  1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Write down the domain of \(f^{-1}\). [1]
Edexcel C3 Q37
4 marks Moderate -0.8
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]
Edexcel FP1 Q26
5 marks Standard +0.3
  1. Express \(\frac{6x + 10}{x + 3}\) in the form \(p + \frac{q}{x + 3}\), where \(p\) and \(q\) are integers to be found. [1]
The sequence of real numbers \(u_1, u_2, u_3, ...\) is such that \(u_1 = 5.2\) and \(u_{n+1} = \frac{6u_n + 10}{u_n + 3}\).
  1. Prove by induction that \(u_n > 5\), for \(n \in \mathbb{Z}^+\). [4]
Edexcel FP2 2008 June Q2
Standard +0.3
  1. Simplify the expression \(\frac{(x + 3)(x + 9)}{x - 1} - (3x - 5)\), giving your answer in the form \(\frac{a(x + b)(x + c)}{x - 1}\), where \(a\), \(b\) and \(c\) are integers. (4)
  2. Hence, or otherwise, solve the inequality \(\frac{(x + 3)(x + 9)}{x - 1} > 3x - 5\) (4)(Total 8 marks)
Edexcel FP2 Q17
5 marks Standard +0.8
  1. Express as a simplified fraction \(\frac{1}{(r-1)^2} - \frac{1}{r^2}\). [2]
  2. Prove, by the method of differences, that $$\sum_{r=2}^{n} \frac{2r-1}{r^2(r-1)^2} = 1 - \frac{1}{n^2}.$$ [3]
Edexcel FP2 Q18
6 marks Standard +0.8
Solve the inequality \(\frac{1}{2x + 1} > \frac{x}{3x - 2}\). [6]
OCR MEI C1 2006 January Q5
4 marks Moderate -0.8
Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]
OCR MEI C1 2011 June Q8
4 marks Moderate -0.8
Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]
OCR MEI C1 2012 June Q4
3 marks Easy -1.2
Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]
Edexcel C1 Q3
5 marks Moderate -0.3
Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]
OCR MEI C1 Q10
3 marks Easy -1.2
Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]
OCR MEI C1 Q3
3 marks Moderate -0.8
Factorise and hence simplify \(\frac{3x^2 - 7x + 4}{x^2 - 1}\). [3]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Simplify \(3a^3b \times 4(ab)^2\). [2]
  2. Factorise \(x^2 - 4\) and \(x^2 - 5x + 6\). Hence express \(\frac{x^2 - 4}{x^2 - 5x + 6}\) as a fraction in its simplest form. [3]
Edexcel C2 Q2
5 marks Moderate -0.3
Express \(\frac{y + 3}{(y + 1)(y + 2)} - \frac{y + 1}{(y + 2)(y + 3)}\) as a single fraction in its simplest form. [5]
Edexcel C2 Q4
7 marks Standard +0.3
Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]
OCR C2 Specimen Q9
11 marks Standard +0.3
The cubic polynomial \(x^3 + ax^2 + bx - 6\) is denoted by f\((x)\).
  1. The remainder when f\((x)\) is divided by \((x - 2)\) is equal to the remainder when f\((x)\) is divided by \((x + 2)\). Show that \(b = -4\). [3]
  2. Given also that \((x - 1)\) is a factor of f\((x)\), find the value of \(a\). [2]
  3. With these values of \(a\) and \(b\), express f\((x)\) as a product of a linear factor and a quadratic factor. [3]
  4. Hence determine the number of real roots of the equation f\((x) = 0\), explaining your reasoning. [3]
Edexcel C2 Q7
9 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$
  1. Solve the equation \(\text{f}(x) = 0\). [3]
  2. Find \(\int \text{f}(x) \, dx\). [3]
  3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]
Edexcel C3 Q1
4 marks Moderate -0.5
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]
Edexcel C3 Q5
7 marks Standard +0.3
Express \(\frac{3}{x^2 + 2x} + \frac{x - 4}{x^2 - 4}\) as a single fraction in its simplest form. [7]
Edexcel C3 Q7
10 marks Moderate -0.3
$$f(x) = \frac{2}{x - 1} - \frac{6}{(x - 1)(2x + 1)}, \quad x > 1$$
  1. Prove that f(x) = \(\frac{4}{2x + 1}\). [4]
  2. Find the range of f. [2]
  3. Find \(f^{-1}(x)\). [3]
  4. Find the range of \(f^{-1}(x)\). [1]
Edexcel C3 Q1
9 marks Moderate -0.3
The function f is given by $$f : x \alpha \frac{x}{x^2-1} - \frac{1}{x+1}, \quad x > 1.$$
  1. Show that \(\text{f}(x) = \frac{1}{(x-1)(x+1)}\). [3]
  2. Find the range of f. [2]
The function g is given by $$g : x \alpha \frac{2}{x}, \quad x > 0.$$
  1. Solve gf(x) = 70. [4]
Edexcel C3 Q2
5 marks Moderate -0.3
Express \(\frac{y+3}{(y+1)(y+2)} - \frac{y+1}{(y+2)(y+3)}\) as a single fraction in its simplest form. [5]
Edexcel C3 Q2
6 marks Standard +0.3
Express \(\frac{x}{(x+1)(x+3)} + \frac{x+12}{x^2-9}\) as a single fraction in its simplest form. [6]