Question

How do you factor $6{x^3} + 1$?

Answer 1

Hint: We will just replace a by $\sqrt[3]{6}x$ and b by 1 in the formula given by the expression: ${a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$. Thus, we will have the required answer.

Complete step by step solution:
We are given that we are required to factor $6{x^3} + 1$.
We know that we have a formula given by the following expression with us:-
$ \Rightarrow {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
Replacing a by $\sqrt[3]{6}x$ and b by 1 in the formula given in the above expression, we will then obtain the following expression with us:-
$ \Rightarrow {\left( {\sqrt[3]{6}x} \right)^3} + {1^3} = \left( {\sqrt[3]{6}x + 1} \right)\left\{ {{{\left( {\sqrt[3]{6}x} \right)}^2} - \left( {\sqrt[3]{6}x} \right)\left( 1 \right) + {1^2}} \right\}$
Simplifying the left hand side of the above equation, we will then obtain the following expression with us:-
$ \Rightarrow 6{x^3} + 1 = \left( {\sqrt[3]{6}x + 1} \right)\left\{ {{{\left( {\sqrt[3]{6}x} \right)}^2} - \left( {\sqrt[3]{6}x} \right)\left( 1 \right) + {1^2}} \right\}$
Simplifying the calculations in the curly brackets on the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow 6{x^3} + 1 = \left( {{6^{\dfrac{1}{3}}}x + 1} \right)\left( {{6^{\dfrac{2}{3}}}{x^2} - {6^{\dfrac{1}{3}}}x + 1} \right)$
Thus, we have the required factors.

Note:-
The students must commit to memory the following formula given by the expression written below:-
$ \Rightarrow {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
We can also verify this formula by opening the right hand side and expanding it as follows:-
Right hand side = $\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)$
We can write this as following expression:-
Right hand side = $a\left( {{a^2} - ab + {b^2}} \right) + b\left( {{a^2} - ab + {b^2}} \right)$
[This is done by using the fact that (u + v) (x + y + z) = u (x + y + z) + v (x + y + z)]
Simplifying it further, we will then obtain the following equation with us:-
Right hand side = ${a^3} - {a^2}b + a{b^2} + {a^2}b - a{b^2} + {b^3}$
[This is done by using the fact that d (e + f + g) = de + df + dg which is known as Distributive Property]
Clubbing the like terms, we will then obtain the following equation with us:-
Right hand side = ${a^3} + {b^3}$
Thus, we have obtained that the left hand side is equal to the right hand side.
Thus, we have verified the formula.