Question

Factorize $2{{x}^{3}}+54$ .

Answer 1

Hint: At first, we take $2$ common from the expression and then express $27$ as ${{3}^{3}}$ and get $2\left[ {{x}^{3}}+{{3}^{3}} \right]$ . Then, we see that $x=-3$ is a root of the polynomial and so, $x+3$ is a factor. Dividing $2\left[ {{x}^{3}}+{{3}^{3}} \right]$ by $x+3$ , we get $2\left( x+3 \right)\left( {{x}^{2}}-3x+9 \right)$ which is the required result.

Complete step by step answer:
The expression that we need to factorize in this problem is,
$2{{x}^{3}}+54$
Now, $54$ can be written as a product of two factors $2$ and $27$ . This will be $2\times 27$ . So, replacing $54$ in the expression with $2\times 27$ , we get the expression as,
$\Rightarrow 2{{x}^{3}}+\left( 2\times 27 \right)$
We know the distributive property which says that an expression of the form $a\left( b+c \right)$ can be written as $ab+ac$ by multiplying a within the brackets. In a reverse manner, we can say that if we are given an expression of the form $ab+ac$ , we can write it as $a\left( b+c \right)$ by taking a common form of the terms. Comparing this with our expression $2{{x}^{3}}+\left( 2\times 27 \right)$ , we can see that $2$ can be taken from the two terms. The expression thus becomes,
$\Rightarrow 2\left[ {{x}^{3}}+27 \right]$
Now, $27$ is a perfect cube. It is a cube of $3$ . So, replacing $27$ with ${{3}^{3}}$ , we get,
$\Rightarrow 2\left[ {{x}^{3}}+{{3}^{3}} \right]$
Now, if we observe carefully, we can see that if we put $x=-3$ , the expression turns out to be zero. This means that $x=-3$ is a root of the expression and $x+3$ is a factor of it. So, dividing the expression ${{x}^{3}}+{{3}^{3}}$ by $x+3$ , we get,
\[\begin{align}
& ~~~~~~~~~{{x}^{2}}-3x+9 \\
& x+3\left| \!{\overline {\,
\begin{align}
& {{x}^{3}}~~~~~~~~~~~~+{{3}^{3}} \\
& {{x}^{3}}+3{{x}^{2}} \\
\end{align} \,}} \right. \\
& x+3\left| \!{\overline {\,
\begin{align}
& -3{{x}^{2}}~~~~~~~~~~~~+27 \\
& -3{{x}^{2}}-9x \\
\end{align} \,}} \right. \\
& x+3\left| \!{\overline {\,
\begin{align}
& 9x+27 \\
& 9x+27 \\
\end{align} \,}} \right. \\
& ~~~~~~~~~~~~\overline{0~~~~~~~~~~~~} \\
\end{align}\]
Thus, we can conclude that $2{{x}^{3}}+54$ upon factorization gives $2\left( x+3 \right)\left( {{x}^{2}}-3x+9 \right)$ .

Note: We can solve this problem in another way. After rewriting the expression as $2\left[ {{x}^{3}}+{{3}^{3}} \right]$ , we can apply the formula ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$ and get the required factorised output.