Manipulating and factorising polynomialsEdexcel International A Level Maths: Subtopic test

1. 1
   The polynomial $f(x)=x^3+4x^2+3x$ is defined for real $x$.

   (a)

   Which of the following is $f(x)$ fully factorised?

   [1 mark]

   • A$x(x+4)(x+3)$
   • B$x(x+1)(x+3)$
   • C$(x+1)(x+3)(x+4)$
   • D$x^2(x+4)+3x$

   (b)

   What is the value of $f(-2)$?

   [1 mark]

   • A$14$
   • B$18$
   • C$-30$
   • D$2$

   (c)

   Solve $f(x)=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g(x)=(2x-1)(x+3)^2$ is defined for real $x$.

   (a)

   Which of the following is the coefficient of $x^2$ when $g(x)$ is expanded and simplified?

   [1 mark]

   • A$13$
   • B$12$
   • C$11$
   • D$6$

   (b)

   Which of the following is the constant term when $g(x)$ is expanded?

   [1 mark]

   • A$-9$
   • B$9$
   • C$-3$
   • D$3$

   (c)

   Show that $g(x)=2x^3+11x^2+12x-9$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The polynomials $h(x)=x^3-9x$ and $k(x)=x^2-x-12$ are defined for real $x$.

   (a)

   Factorise $h(x)$ completely.

   [3 marks]

   (b)

   Expand and simplify $h(x)-x\,k(x)$, and hence write it in fully factorised form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A closed box has a square base of side $(x+2)$ cm and height $(2x-1)$ cm, where $x>1$.

   (a)

   (i) Show that the volume of the box is $V=2x^3+7x^2+4x-4$ cm$^3$.
   (ii) Show that the total surface area of the box is $10x(x+2)$ cm$^2$.

   [6 marks]

   (b)

   The box is filled with sand to a depth of $x$ cm.
   (i) Find, in terms of $x$, the volume of the empty space above the sand, giving your answer as an expanded polynomial.
   (ii) Hence show that the volume of the sand is $x(x+2)^2$ cm$^3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions