Simplifying rational expressionsEdexcel International A Level Maths: Subtopic test

1. 1
   Let $f(x)=\dfrac{x^2-9}{x^2+x-12}$.

   (a)

   Which expression is equal to $f(x)$ for all values of $x$ for which $f(x)$ is defined?

   [1 mark]

   • A$\frac{x-3}{x+4}$
   • B$\frac{x+3}{x+4}$
   • C$\frac{-9}{x-12}$
   • D$\frac{x+3}{x-4}$

   (b)

   For which values of $x$ is the original expression $f(x)$ undefined?

   [1 mark]

   • A$x=-4$ only
   • B$x=3$ only
   • C$x=3$ and $x=-3$
   • D$x=3$ and $x=-4$

   (c)

   Solve $f(x)=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $h(x)=\dfrac{x^3+1}{x^2-1}$.

   (a)

   Which is a complete factorisation of $x^3+1$?

   [1 mark]

   • A$(x-1)(x^2+x+1)$
   • B$(x+1)^3$
   • C$(x+1)(x^2-x+1)$
   • D$(x+1)(x^2+x+1)$

   (b)

   Which expression equals $h(x)$ for all $x$ for which $h(x)$ is defined?

   [1 mark]

   • A$\frac{x^2-x+1}{x-1}$
   • B$\frac{x^2+x+1}{x-1}$
   • C$x^2-x+1$
   • D$\frac{x^2-x+1}{x+1}$

   (c)

   Hence find constants $a$, $b$ and $c$ such that $h(x)\equiv ax+b+\dfrac{c}{x-1}$ for $x\neq\pm1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $F(x)=\dfrac{2x^3-3x^2-11x+6}{x^2-x-6}$, where $x\in\mathbb{R}$, $x\neq 3$, $x\neq -2$.

   (a)

   Simplify $F(x)$ fully, showing your factorisation of the numerator.

   [3 marks]

   (b)

   Explain why the graph of $y=F(x)$ is the straight line $y=2x-1$ with two points missing, and find the coordinates of the two missing points.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\dfrac{2x^3+3x^2-8x+1}{x^2-x-2}$ and $g(x)=\dfrac{x^3-8}{x^2-4}$.

   (a)

   (i) Use algebraic division to find constants $A$, $B$, $C$ and $D$ such that $f(x)\equiv Ax+B+\dfrac{Cx+D}{x^2-x-2}$.
   (ii) State the values of $x$ for which $f(x)$ is not defined.
   (iii) Explain whether $\dfrac{Cx+D}{x^2-x-2}$ can be simplified further by cancelling.

   [6 marks]

   (b)

   (i) Simplify $g(x)$ by factorising.
   (ii) Hence write $g(x)$ in the form $x+\dfrac{p}{x+q}$.
   (iii) Comment on $g(2)$ and on the value of your simplified expression at $x=2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions