Remainder and factor theorems questions with answers

Questions

1. Question 1Remainder: divide by (x − a)

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

   Show the worked solution

   1. Step 1: Remainder theorem: the remainder is f(the value of x that makes the divisor 0)$x - 2 = 0$ when $x = 2$
   2. Step 2: Substitute that value of x into f(x), work out each term, then add$f(2) = 3(2)^3 - (2)^2 + 2(2) - 5$$= 24 - 4 + 4 - 5$$= 19$The remainder is $19$.
2. Question 2Remainder: divide by (bx − a)

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

   Show the worked solution

   1. Step 1: Remainder theorem: the remainder is f(the value of x that makes the divisor 0)$2x - 3 = 0$ when $x = \dfrac{3}{2}$
   2. Step 2: Substitute that value of x into f(x), work out each term, then add$f(\dfrac{3}{2}) = (\dfrac{3}{2})^3 - 2(\dfrac{3}{2})^2 + 5(\dfrac{3}{2}) - 3$$= \dfrac{27}{8} + \dfrac{-9}{2} + \dfrac{15}{2} - 3$$= \dfrac{27}{8}$The remainder is $\dfrac{27}{8}$.
3. Question 3Find an unknown coefficient

   Given that $x + 1$ is a factor of $3x^3 + kx^2 + 2x - 5$, find the value of $k$.

   Show the worked solution

   1. Step 1: Factor theorem: (x − a) is a factor, so f(a) = 0$f(-1) = 0$
   2. Step 2: Substitute that value of x, work out each term, then solve for k$3(-1)^3 + k(-1)^2 + 2(-1) - 5 = 0$$k - 3 - 2 - 5 = 0$$k - 10 = 0$$k = 10$
4. Question 4Remainder on dividing by a quadratic

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

   Show the worked solution

   1. Step 1: Dividing by a quadratic leaves a remainder of degree at most 1: call it ax + b$f(x) = (x - 2)(x + 1)Q(x) + ax + b$
   2. Step 2: Factorise the divisor, then find f(2) and f(-1)$f(2) = 33$$f(-1) = 0$
   3. Step 3: Put x = 2 and x = -1: the quadratic is 0, leaving two equations$2a + b = 33$$-a + b = 0$
   4. Step 4: Solve the simultaneous equations$a = 11$$b = 11$
   5. Step 5: Write the remainder$11x + 11$