The factor theoremAQA A-Level Maths: Subtopic test

1. 1
   Let $f(x)=2x^3-x^2-7x+6$.

   (a)

   Which of these is a factor of $f(x)$?

   [1 mark]

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

   (b)

   Given that $(2x-3)$ is a factor of $f(x)$, which statement must be true?

   [1 mark]

   • A$f\left(\frac32\right)=0$
   • B$f(3)=0$
   • C$f\left(-\frac32\right)=0$
   • D$f\left(\frac23\right)=0$

   (c)

   Given that $(x-1)$ and $(2x-3)$ are factors of $f(x)$, factorise $f(x)$ completely.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The polynomial $p(x)=x^3+kx^2-4x+12$ has $(x-2)$ as a factor, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$k=3$
   • B$k=-6$
   • C$k=-3$
   • D$k=-12$

   (b)

   Using your value of $k$, which of these is NOT a factor of $p(x)$?

   [1 mark]

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

   (c)

   Using your value of $k$, solve $p(x)=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The cubic $g(x)=2x^3+ax^2+bx-12$, where $a$ and $b$ are constants, has factors $(x-2)$ and $(x+3)$.

   (a)

   Use the factor theorem to show that $2a+b=-2$ and $3a-b=22$.

   [3 marks]

   (b)

   Hence find the values of $a$ and $b$, and factorise $g(x)$ completely.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=x^4-3x^3-7x^2+15x+18$ and $g(x)=2x^3+ax^2-x+b$, where $a$ and $b$ are constants.

   (a)

   (i) Show that $(x+1)$ and $(x-3)$ are factors of $f(x)$.
   (ii) Hence factorise $f(x)$ completely.
   (iii) State the solutions of $f(x)=0$, identifying any repeated solution.

   [6 marks]

   (b)

   The cubic $g(x)$ has $(x-1)$ and $(2x+1)$ as factors. Find the values of $a$ and $b$, and hence solve $g(x)=0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions