Question

How do you factor the polynomial $3{{x}^{3}}-9{{x}^{2}}-54x+120$ ?

Answer 1

Hint: In this question, we have to find the factors of a polynomial. Thus, we will use the hit and trial method, long division method and splitting the middle term method to get the solution. Therefore, to solve this problem, we will first find a root of the given polynomial by letting different values of x, whichever value gives the answer as 0, implies the value is the root of the polynomial. After that we will divide the polynomial by the root, to get a quotient. Then, we will solve the quotient using splitting the middle term method, to get the required solution to the problem.

Complete step-by-step solution:
According to the question, we have to find the factors of the polynomial.
The given polynomial to us is $3{{x}^{3}}-9{{x}^{2}}-54x+120$ ----------- (1)
Now, let $x=2$ and put this value I the equation (1), we get
$\begin{align}
  & \Rightarrow 3{{(2)}^{3}}-9.{{(2)}^{2}}-54.(2)+120 \\
 & \Rightarrow 24-36-108+120 \\
 & \Rightarrow 144-144 \\
\end{align}$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow 0$
This implies that $x=2$ is the solution of the given polynomial.
Therefore, one of the roots of the given polynomial is $(x-2)$ ---- (2)
Now, we will divide equation (1) by equation (2) by long division method, we get
\[x-2\overset{3{{x}^{2}}-3x-60}{\overline{\left){\begin{align}
  & 3{{x}^{3}}-9{{x}^{2}}-54x+120 \\
 & \underline{{}_{-}3{{x}^{3}}-{}_{+}6{{x}^{2}}} \\
 & \text{ }-3{{x}^{2}}-54x+120 \\
 & \text{ }\underline{{}_{+}-3{{x}^{2}}+{}_{-}6x\text{ }} \\
 & \text{ }-60x+120 \\
 & \text{ }\underline{{}_{+}-60x+{}_{-}120} \\
 & \text{ }\underline{\text{ }0} \\
 & \text{ } \\
\end{align}}\right.}}\]
Therefore, we get the remainder equals 0 and the quotient is equal to $3{{x}^{2}}-3x-60$ .
Now, we will use the splitting the middle term method in the quotient, which is
$3{{x}^{2}}-3x-60$
So, we will split the middle term as the sum of -15x and 12x, because $a.c=\left( -15 \right)12=-180$ and $a+c=-15+12=-3$
Therefore, we get
$\Rightarrow 3{{x}^{2}}-15x+12x-60$
Now, we take 3x common from the first two terms and take common 12 from the last two terms, we get
$\Rightarrow 3x\left( x-5 \right)+12\left( x-5 \right)$
Now, take (x-5) common from the above equation, we get
$\Rightarrow \left( x-5 \right)\left( 3x+12 \right)$ ------- (3)
From equation (2) and (3), we get
$\left( x-2 \right)\left( x-5 \right)\left( 3x+12 \right)$
Therefore, for the equation $3{{x}^{3}}-9{{x}^{2}}-54x+120$ , its factors are $\left( x-2 \right)\left( x-5 \right)\left( 3x+12 \right)$.

Note: While solving this problem, do mention all the formulas and the methods you are using to solve the problem. One of the alternative methods to solve the quadratic equation is you can use the discriminant formula or the hit and trial method to get the solution.