Polynomial manipulation and algebraic divisionAQA A-Level Maths: Subtopic test

1. 1
   Let $p(x)=2x^3-x^2-13x-6$ and $q(x)=x^3+3x^2-5x+4$.

   (a)

   Which expression is equal to $p(x)-q(x)$?

   [1 mark]

   • A$3x^3+2x^2-18x-2$
   • B$x^3-4x^2-18x-10$
   • C$x^3-4x^2-8x-2$
   • D$x^3-4x^2-8x-10$

   (b)

   What is the coefficient of $x^3$ in the expansion of $(x-1)\,q(x)$?

   [1 mark]

   • A$3$
   • B$4$
   • C$2$
   • D$-1$

   (c)

   Given that $(x+2)$ is a factor of $p(x)$, factorise $p(x)$ completely.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The expressions $A=x^4-81$ and $B=6x^2+7x-3$ are to be factorised.

   (a)

   Which is the complete factorisation of $A$?

   [1 mark]

   • A$(x^2+9)(x-3)(x+3)$
   • B$(x^2-9)(x^2+9)$
   • C$(x-3)^2(x+3)^2$
   • D$(x-3)^4$

   (b)

   Which is the factorisation of $B$?

   [1 mark]

   • A$(3x+1)(2x-3)$
   • B$(3x-1)(2x+3)$
   • C$(6x-1)(x+3)$
   • D$(3x-1)(2x-3)$

   (c)

   Factorise completely $x^3+3x^2-4x-12$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=3x^3+2x^2-19x+6$.

   (a)

   Use algebraic division to show that $f(x)=(x-2)(3x^2+8x-3)$.

   [3 marks]

   (b)

   Hence factorise $f(x)$ completely and solve $f(x)=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=2x^3+7x^2-5x-4$ and $g(x)=(2x-1)^2(x+3)-(x+3)^3$.

   (a)

   (i) Use algebraic division to show that $f(x)=(x-1)(2x^2+9x+4)$.
   (ii) Hence factorise $f(x)$ completely.
   (iii) Hence solve $f(x)=0$.

   [6 marks]

   (b)

   Show that $g(x)=3x^3-x^2-38x-24$, and factorise $g(x)$ completely.

   [6 marks]

   Total for question 4: 12 marks

End of questions