Factor Theorem and Remainder TheoremEdexcel International A Level Maths: Subtopic test

1. 1
   Let $f(x)=x^3-7x+6$.

   (a)

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

   [1 mark]

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

   (b)

   Find the remainder when $f(x)$ is divided by $(x-3)$.

   [1 mark]

   • A$6$
   • B$12$
   • C$-6$
   • D$-12$

   (c)

   Show that $(x+3)$ is a factor of $f(x)$.

   [2 marks]

   Total for question 1: 4 marks

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

   (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)

   Find the remainder when $g(x)$ is divided by $(x+1)$.

   [1 mark]

   • A$6$
   • B$20$
   • C$18$
   • D$-4$

   (c)

   Show that $(x+3)$ is a factor of $g(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x)=6x^3+11x^2-3x-2$.

   (a)

   Show that $(x+2)$ is a factor of $f(x)$, and find the quadratic $q(x)$ such that $f(x)=(x+2)q(x)$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The polynomial $p(x)=2x^3+ax^2+bx-6$, where $a$ and $b$ are constants, has $(x-2)$ as a factor and leaves remainder $-12$ when divided by $(x+1)$.

   (a)

   Find the values of $a$ and $b$.

   [6 marks]

   (b)

   Given that $a=-3$ and $b=1$, show that $p(x)=0$ has exactly one real root.

   [6 marks]

   Total for question 4: 12 marks

End of questions