Question

How do you factor $27{x^3} + 1?$

Answer 1

Hint: As the general expression of the equation having cubic power is $f\left( x \right) = a{x^3} + {b^2} + c{x^1} + d$, thus we can write coefficients $a,b,$ and $c$ where $d$ is constant.
As, above expression can be written in form of cubic expression like,
${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$, so with the help of the mentioned identity we can determine the value of given expression.

Complete step-by-step solution:
As per data given in the question,
We have to determine the value of expression $27{x^3} + 1$
As we know that the above equation can be solved as given below.
So, we can write the above equation as
As we know that the value of cube of 3 is 27,
So, we can rewrite $27{x^3}$ as ${\left( {3x} \right)^3}$
As, value of cube of 1 is always 1
So, we can also write $1$ as ${\left( 1 \right)^3}$
After putting the value in above expression,
We will get,
The value of equation $27{x^3} + 1$ can be written as ${\left( {3x} \right)^3} + {\left( 1 \right)^3}$
So, by using the identity formula ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
So, the equation ${\left( {3x} \right)^3} + {\left( 1 \right)^3}$
After putting the values i.e. a = 3x and b = 1,
We will get,
$\left( {3x + 1} \right).\left[ {{{\left( {3x} \right)}^2} - \left( {3x} \right).\left( 1 \right) + {{\left( 1 \right)}^2}} \right]$
The above equation is done by cubes formula.
$\left( {3x + 1} \right)\left( {9{x^2} - 3x + 1} \right)$

Therefore $\left( {3x + 1} \right)\left( {9{x^2} - 3x + 1} \right)$ is the required answer.

Additional Information: So, the equation is solved in Cubic formula
$\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} + {a^2}b + a{b^2} - {a^2}b - a{b^2} - {b^2}$
So, now further solving the equation we have
$ = {a^3} + {a^2}b - {a^2}b + a{b^2} - a{b^2} - {b^3}$
Further solving the equation we get.
$ = {a^3} - {b^3}$
So, we had done
${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$
Thus above is the proof of the cubic formula.

Note:
As we know that the above equation are solved by cubic equation so one of the example we have $2{x^2} - 8$
We can further value it.
$2{x^2} - 8$ We can write it as $2{x^2}\left( {2 \times 2 \times 2} \right) = 2{x^2} - \left( {24} \right)$
${x^2} - 4$ cam be written as ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
Further solving we get.
Therefore
$2\left( {{x^2} - 4} \right) = 2\left( {x + 2} \right).\left( {x - 2} \right)$