Question

How do you factor $6{{x}^{3}}-3{{x}^{2}}+9$ ?

Answer 1

Hint: In this question, we have to find the factors of a cubic polynomial. 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 given polynomial. After that we will divide the polynomial by the root, to get a quotient in the form of a quadratic equation, 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 cubic polynomial.
The given polynomial to us is $6{{x}^{3}}-3{{x}^{2}}+9$ ----------- (1)
Now, let $x=-1$ and put this value in the equation (1), we get
$\begin{align}
  & \Rightarrow 6{{\left( -1 \right)}^{3}}-3.{{\left( -1 \right)}^{2}}+9 \\
 & \Rightarrow -6-3+9 \\
 & \Rightarrow -9+9 \\
\end{align}$
As we know, the same terms with opposite signs cancel out each other, therefore we get
$\Rightarrow 0$
This implies that $x=-1$ is the solution of the given polynomial.
Therefore, one of the roots of the given polynomial is $(x+1)$ ---- (2)
Now, we will divide equation (1) by equation (2) by long division method, we get
$\begin{align}
  & \text{ 6}{{\text{x}}^{2}}-9x+9 \\
 & \text{x+1}\left| \!{\overline {\,
 6{{x}^{3}}-3{{x}^{2}}+0x+9 \,}} \right. \\
 & \;\;\;\;\;\;\;\;\;\; \underline{{}_{-}\text{6}{{\text{x}}^{3}}+{}_{-}6{{x}^{2}}} \\
 & \;\;\;\;\;\;\;\;\;\; -9{{\text{x}}^{2}}\text{+0x+9 } \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\; \underline{{}_{+}\text{-9}{{\text{x}}^{2}}-{}_{+}\text{9x }}\text{ } \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 9x+9 \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \underline{{}_{-}\text{9x+}{}_{-}9} \\
 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text{ 0 }} \\
\end{align}$
Therefore, we get the remainder equals 0 and the quotient is equal to $6{{x}^{2}}-9x+9$ .
Now, we will use the splitting the middle term method in the quotient, which is
$\Rightarrow 6{{x}^{2}}-9x+9$ ------ (3)
Since, we cannot further find the factors of the above quadratic equation.
Thus, from equation (2) and (3), we get
$\Rightarrow \left( x+1 \right)\left( 6{{x}^{2}}-9x+9 \right)$
Therefore, for the equation $6{{x}^{3}}-3{{x}^{2}}+9$ , its factors are $\left( x+1 \right)\left( 6{{x}^{2}}-9x+9 \right)$.

Note: While solving this problem, do mention all the formulas and the methods you are using to avoid confusion and mathematical errors. Do not forget that we cannot solve this $\left( 6{{x}^{2}}-9x+9 \right)$ equation further because we get the complex roots of the equation.