Polynomials: practice questions

6 questions · 12 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   Find the remainder when $x^3 - 2x^2 + 5$ is divided by $x + 1$.

   A

   $4$

   B

   $8$

   C

   $2$

   D

   $-2$

2. Question 2Multiple choice · 1 mark

   If $x - 2$ is a factor of $x^3 + kx - 6$, find $k$.

   A

   $-1$

   B

   $1$

   C

   $3$

   D

   $-7$

3. Question 3Multiple choice · 1 mark

   Find the remainder when $4x^2 - 3x + 1$ is divided by $2x - 1$.

   A

   $0$

   B

   $\dfrac{3}{2}$

   C

   $2$

   D

   $\dfrac{1}{2}$

4. Question 4Multiple choice · 1 mark

   Factorise $x^3 - 7x + 6$.

   A

   $(x + 1)(x + 2)(x - 3)$

   B

   $(x - 1)(x - 2)(x + 3)$

   C

   $(x - 1)(x + 2)(x - 3)$

   D

   $(x - 1)^2(x + 6)$

5. Question 55 marks

   Let $f(x) = 2x^3 + ax^2 + bx - 6$. When $f(x)$ is divided by $x - 1$, the remainder is $-6$. Also, $x + 2$ is a factor of $f(x)$.

   (a) Find $a$ and $b$.

   (b) Hence factorise $f(x)$ completely.

   Show worked solution

   (a) $f(1) = -6$: $2 + a + b - 6 = -6$, so $a + b = -2$ ... (1) 1M

   $f(-2) = 0$: $-16 + 4a - 2b - 6 = 0$, so $2a - b = 11$ ... (2) 1M

   Solving, $a = 3$ and $b = -5$. 1A

   (b) $f(x) = 2x^3 + 3x^2 - 5x - 6 = (x + 2)(2x^2 - x - 3)$ 1M

   $= (x + 2)(2x - 3)(x + 1)$ 1A

6. Question 63 marks

   Simplify $\dfrac{1}{x - 1} - \dfrac{2}{x^2 - 1}$.

   Show worked solution

   $x^2 - 1 = (x - 1)(x + 1)$, so the L.C.M. of the denominators is $(x - 1)(x + 1)$. 1M

   $$\frac{1}{x - 1} - \frac{2}{x^2 - 1} = \frac{(x + 1) - 2}{(x - 1)(x + 1)} \qquad \text{1M}$$ $$= \frac{1}{x + 1} \qquad \text{1A}$$