P2: Algebra and functionsEdexcel International A Level Maths: Topic test

1. 1
   The polynomial $f(x)=4x^3-5x^2+3x-7$ is divided by $(x-2)$.

   (a)

   What is the remainder?

   [1 mark]

   • A$11$
   • B$-65$
   • C$9$
   • D$-5$

   (b)

   What is the quotient?

   [1 mark]

   • A$4x^2-5x+3$
   • B$4x^2-13x+29$
   • C$4x^2+3x+9$
   • D$4x^2+3x+11$

   (c)

   Find the remainder when $f(x)$ is divided by $(2x+1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The polynomial $g(x)=2x^3+ax^2-11x+b$, where $a$ and $b$ are constants, has $(x-3)$ as a factor. When $g(x)$ is divided by $(x+1)$ the remainder is $12$.

   (a)

   Which equation in $a$ and $b$ follows from the fact that $(x-3)$ is a factor?

   [1 mark]

   • A$9a+b-21=0$
   • B$9a+b+21=0$
   • C$3a+b+21=0$
   • D$9a+b-15=0$

   (b)

   Find the value of $b$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$9$
   • D$6$

   (c)

   Using the values of $a$ and $b$, find the remainder when $g(x)$ is divided by $(x-2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $f(x)=6x^3-5x^2-17x+6$.

   (a)

   Find the quotient and the remainder when $f(x)$ is divided by $(2x-1)$.

   [3 marks]

   (b)

   Show that $(x-2)$ is a factor of $f(x)$, and hence solve $f(x)=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The polynomial $p(x)=3x^3+ax^2+bx-4$, where $a$ and $b$ are constants, has $(x+2)$ as a factor. When $p(x)$ is divided by $(3x-1)$ the remainder is $-\dfrac{70}{9}$.

   (a)

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

   [6 marks]

   (b)

   Given that $a=1$ and $b=-12$: (i) factorise $p(x)$ completely and hence solve $p(x)=0$ [4 marks]; (ii) find the quotient and the remainder when $p(x)$ is divided by $(x-1)$ [2 marks].

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $f(x)=2x^3-9x^2+x+12$.

   (a)

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

   [1 mark]

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

   (b)

   What is the remainder when $f(x)$ is divided by $(x-2)$?

   [1 mark]

   • A$-6$
   • B$6$
   • C$-42$
   • D$12$

   (c)

   Factorise $f(x)$ completely.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The polynomial $h(x)=8x^3-6x^2+5x-1$ is divided by $(2x-1)$.

   (a)

   What is the quotient?

   [1 mark]

   • A$8x^2-2x+4$
   • B$4x^2-x+1$
   • C$4x^2-x+2$
   • D$4x^2-x+3$

   (b)

   What is the remainder?

   [1 mark]

   • A$-1$
   • B$1$
   • C$-6$
   • D$6$

   (c)

   The polynomial $h(x)+kx$, where $k$ is a constant, leaves remainder $4$ when divided by $(2x-1)$. Find the value of $k$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   $f(x)=x^3-6x^2+kx+10$, where $k$ is a constant.

   (a)

   The remainder when $f(x)$ is divided by $(x-3)$ is $-8$. Find the value of $k$.

   [3 marks]

   (b)

   Given that $k=3$, show that $(x+1)$ is a factor of $f(x)$ and solve $f(x)=0$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The polynomial $f(x)=3x^3+px^2+qx+2$, where $p$ and $q$ are constants, has $(x+2)$ as a factor. When $f(x)$ is divided by $(x+1)$ the remainder is $8$.

   (a)

   Find the values of $p$ and $q$.

   [6 marks]

   (b)

   Given that $p=2$ and $q=-7$: (i) find the quotient and the remainder when $f(x)$ is divided by $(3x-1)$ [3 marks]; (ii) hence factorise $f(x)$ completely and solve $f(x)=0$ [3 marks].

   [6 marks]

   Total for question 8: 12 marks

End of questions