Question

How do you factor $8{x^3} + 27?$

Answer 1

Hint: Here we will first observe the given expression and will convert the given expression in the form of cubes. Use the identity of the algebraic expressions ${a^3} + {b^3}$ and then will simplify for the required answer.

Complete step-by-step answer:
Take the given expression –
$8{x^3} + 27$
We can observe that ${2^3}$ is $8$ and ${3^3} = 27$ and therefore the above expression can be re-written as –
${(2x)^3} + {(3)^3}$
Use the identity of the sum of the cubes as ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$in the above equation.
Here, $a = 2x$ and $b = 3$
\[{(2x)^3} + {(3)^3} = (2x + 3)\left( {{{(2x)}^2} - (2x)(3) + {{(3)}^2}} \right)\]
Simplify the above equation writing the squares of the terms in the bracket on the left.
\[{(2x)^3} + {(3)^3} = (2x + 3)\left( {4{x^2} - 6x + 9} \right)\]
This is the required solution.

Additional Information:
Know the concepts of squares and cubes. Square is the number multiplied itself and cube it the number multiplied thrice. Square is the product of same number twice such as ${n^2} = n \times n$ for Example square of $2$ is ${2^2} = 2 \times 2$ simplified form of squared number is ${2^2} = 2 \times 2 = 4$ and square-root is denoted by $\sqrt {{n^2}} = \sqrt {n \times n} $ For Example: $\sqrt {{2^2}} = \sqrt 4 = 2$ Similarly cube is the product of same number three times such as ${n^3} = n \times n \times n$ for Example cube of $2$ is ${2^3} = 2 \times 2 \times 2$ simplified form of cubed number is ${2^3} = 2 \times 2 \times 2 = 8$. and cube-root is denoted by $\sqrt[3]{{{n^3}}} = \sqrt {n \times n \times n} = n$ For Example: $\sqrt[3]{8} = \sqrt[3]{{{2^3}}} = 2$ Do not be confused in square and square-root similarly cubes and cube-root, know the concepts properly and apply accordingly.

Note: To find the factors of the algebraic expressions remember the identities relating to the sum and difference of cubes and sum and difference of squares formulas for an accurate and an efficient solution.