Polynomials, factor theorem and algebraic divisionEdexcel A-Level Maths: Subtopic test

1. 1
   The polynomial $f(x)=x^3+3x^2-4$.

   (a)

   Which of the following is a factor of $f(x)$?

   [1 mark]

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

   (b)

   Which of the following is the complete factorisation of $f(x)$?

   [1 mark]

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

   (c)

   Hence solve $f(x)=0$, stating which root is repeated.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The polynomial $g(x)=2x^3+x^2-13x+6$.

   (a)

   Which of the following is a factor of $g(x)$?

   [1 mark]

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

   (b)

   $g(x)$ is divided by $(x-2)$, which is a factor. What is the quotient?

   [1 mark]

   • A$2x^2-5x-3$
   • B$2x^2+5x+3$
   • C$2x^2+x-13$
   • D$2x^2+5x-3$

   (c)

   Given that $(x-2)$ is a factor, factorise $g(x)$ completely.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The polynomial $p(x)=2x^3+ax^2-x+6$, where $a$ is a constant, has a factor $(x-2)$.

   (a)

   Find the value of $a$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cubic polynomial $f(x)=6x^3+11x^2-x-6$.

   (a)

   (i) Show that $(x+1)$ is a factor of $f(x)$.
   (ii) Hence factorise $f(x)$ completely.
   (iii) State the values of $x$ where the curve $y=f(x)$ meets the $x$-axis.

   [6 marks]

   (b)

   A different cubic is $r(x)=6x^3+11x^2+ax+b$, where $a$ and $b$ are constants. Given that $(x+1)$ and $(x-1)$ are both factors of $r(x)$:
   (i) find the values of $a$ and $b$;
   (ii) hence factorise $r(x)$ completely.

   [6 marks]

   Total for question 4: 12 marks

End of questions