Question

Factorize the given equation:
$2{x^3} - 3{x^2} - 17x + 30 = 0$
$
  (a){\text{ }}\left( {x - 2} \right)\left( {x + 4} \right)\left( {2x - 5} \right) \\
  (b){\text{ }}\left( {x - 2} \right)\left( {x + 2} \right)\left( {2x - 5} \right) \\
  (c){\text{ }}\left( {x - 2} \right)\left( {x + 1} \right)\left( {2x - 5} \right) \\
  (d){\text{ }}\left( {x - 2} \right)\left( {x + 3} \right)\left( {2x - 5} \right) \\
$

Answer 1

Hint – There can be two ways to solve this problem. Firstly is the simple use of hit and trial to match up with the options for value of x. In option, if we have $\left( {x - 2} \right)$ this means that 2 is the value of x which we have to satisfy in the given equation. Second method we will explain in a later ending.

Complete step-by-step answer:
Given equation is
$2{x^3} - 3{x^2} - 17x + 30$

We solve this equation by hit and trial method.

So substitute x = 2 in the equation we have,
$ \Rightarrow 2{\left( 2 \right)^3} - 3{\left( 2 \right)^2} - 17\left( 2 \right) + 30$

Now simplify it we will get,
$ \Rightarrow 16 - 12 - 34 + 30 = 46 - 46 = 0$

So (x – 2) is one of the factors of the equation.

Now substitute x = -3 in the equation we have,
$ \Rightarrow 2{\left( { - 3} \right)^3} - 3{\left( { - 3} \right)^2} - 17\left( { - 3} \right) + 30$

Now simplify it we will get,
$ \Rightarrow 2\left( { - 27} \right) - 3\left( 9 \right) + 51 + 30 = - 54 - 27 + 81 = - 81 + 81 = 0$

So (x + 3) is also one of the factors of the equation.

Now substitute x = $\dfrac{5}{2}$ in the equation we have,
$ \Rightarrow 2{\left( {\dfrac{5}{2}} \right)^3} - 3{\left( {\dfrac{5}{2}} \right)^2} - 17\left( {\dfrac{5}{2}} \right) + 30$

Now simplify it we will get,
$ \Rightarrow 2\left( {\dfrac{{125}}{8}} \right) - 3\left( {\dfrac{{25}}{4}} \right) - \dfrac{{85}}{2} + 30$

Now above equation is also written as
$ \Rightarrow \dfrac{{125}}{4} - \dfrac{{75}}{4} - \dfrac{{170}}{4} + \dfrac{{120}}{4} = \dfrac{{245}}{4} - \dfrac{{245}}{4} = 0$

So (2x – 5) is also one of the factors of the equation.

So the factors of the given equation is
$ \Rightarrow \left( {x - 2} \right)\left( {x + 3} \right)\left( {2x - 5} \right)$

Hence option (d) is correct.

Note – The second method is, use Long division between the given polynomial equation and the root obtained to form the remaining quadratic equation. Then apply middle term splitting for factorization of the obtained quadratic to find the remaining two roots.