Question

Find the value of b for which the polynomial $$2{x^3} + 9{x^2} - x - b$$ is exactly divisible by $$2x + 3$$

Answer 1

Hint: Here polynomial is given, we need to apply the factor theorem method. After putting the value by factor method, we will get equations. After solving the equation we can get the answer.

Complete step-by-step answer:
Since $$2x + 3\;$$is a factor of the polynomial $$p\left( x \right) = 2{x^3} + 9{x^2} - x - b$$
Therefore, by Factor theorem $$p(\dfrac{{ - 3}}{2}) = 0$$
$$\eqalign{
  & \Rightarrow 2(\dfrac{{ - 3}}{2})^3 + 9(\dfrac{{ - 3}}{2})^2 - (\dfrac{{ - 3}}{2}) - b = 0 \cr
  & \Rightarrow \dfrac{{ - 27}}{4} + \dfrac{{81}}{4} + \dfrac{3}{2} - b = 0 \cr
  & \Rightarrow \dfrac{{ - 27 + 81 + 6}}{4} - b = 0 \cr
  & \Rightarrow b = \dfrac{{60}}{4} = 15 \cr
  & \therefore \;b = 15 \cr} $$
Hence the value of b for which the polynomial polynomial $$2{x^3} + 9{x^2} - x - b$$ is exactly divisible by $$2x + 3$$ is 15.

Note: In this type of problem we need to apply the factor theorem, after applying the factor method, we got the equation, after solving the equations i.e. equating LHS and RHS we got the answer. The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros.