How do you factor $27{{g}^{3}}+343$ ?

Hint: Here we have to find the factor of the expression given. Firstly we know that factors of an expression are those terms which divide the expression completely. So we will simplify the expression given by writing the two terms in the expression in cube form. Then we will use the algebraic identity ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$ and simplify our expression further and get our desired answer.


Complete step by step answer:
We have to find the factor of the expression given as follows:
$27{{g}^{3}}+343$…..$\left( 1 \right)$
Now factor means we have to divide the above expression in terms which one product gives back the expression.
As we know $343={{7}^{3}}$ and $27={{3}^{3}}$ using these value in equation (1) we get,
$\Rightarrow {{\left( 3g \right)}^{3}}+{{\left( 7 \right)}^{3}}$…..$\left( 2 \right)$
Next we will use the algebraic identity given below:
${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$
Comparing above identity with equation (2) we get,
$a=3g$ and $b=7$
On substituting the above values in the formula we get,
$\Rightarrow {{\left( 3g \right)}^{3}}+{{\left( 7 \right)}^{3}}=\left( 3g+7 \right)\left( {{\left( 3g \right)}^{2}}-3g\times 7+{{7}^{2}} \right)$
$\Rightarrow {{\left( 3g \right)}^{3}}+{{\left( 7 \right)}^{3}}=\left( 3g+7 \right)\left( 9{{g}^{2}}-21g+49 \right)$
So got the answer as $\left( 3g+7 \right)\left( 9{{g}^{2}}-21g+49 \right)$ .
Hence the two factors of $27{{g}^{3}}+343$ are $\left( 3g+7 \right)\left( 9{{g}^{2}}-21g+49 \right)$ .

Note: Algebraic expression is any mathematical phrase which contains variables and constants along with algebraic operations operating on them. An algebraic identity is that algebraic equation which is true for all values of variables in it. The difference between algebraic equation and expression is that the first one contains an equal sign and the other doesn’t contain an equal sign. As our question has a variable with highest power as $3$ so we used the cube formula for simplifying it. Similarly if the highest power is $2$ we can use the square algebraic identity to simplify our expression. Factors of an expression are those values which on multiplication with an constant or other expression give backs the original expression.