Question

How do you factor \[8{{y}^{3}}+1\]?

Answer 1

Hint: For the given question we are asked to find the factor \[8{{y}^{3}}+1\]. For this question we will bring the question into \[{{a}^{3}}+{{b}^{3}}\] form and we will use the algebraic formula which is \[{{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)\]
Firstly, as mentioned in the above will try and bring the given question into \[{{a}^{3}}+{{b}^{3}}\]form.

Complete step-by-step solution:
So, we get the expression as follows.
\[\Rightarrow {{\left( 2y \right)}^{3}}+{{1}^{3}}\]
So, after bringing the given expression into \[{{a}^{3}}+{{b}^{3}}\]form we will use the basic algebraic formula which is \[{{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)\].
So, the equation after using the formula will be reduced as follows.
\[\Rightarrow {{\left( 2y \right)}^{3}}+{{1}^{3}}={{\left( 2y+1 \right)}^{3}}-3\times 2y\times 1\left( 2y+1 \right)\]
Here we will take the\[\left( 2y+1 \right)\] common in the right hand side of the equation and further simplify the equation as follows.
\[\Rightarrow \left( 2y+1 \right)\left[ {{\left( 2y+1 \right)}^{2}}-6y \right]\]
Here inside the square bracket for square of \[\left( 2y+1 \right)\] we will use the basic formula in algebra which is \[\Rightarrow {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\].
After using the formula we will get the equation as follows.
\[\Rightarrow \left( 2y+1 \right)\left[ 4{{y}^{2}}+4y+1-6y \right]\]
\[\Rightarrow \left( 2y+1 \right)\left[ 4{{y}^{2}}-2y+1 \right]\]
Therefore, the solution to the given question will be \[ \left( 2y+1 \right)\left[ 4{{y}^{2}}-2y+1 \right]\].

Note: Students must be very careful while solving questions of this kind and students must be having good knowledge in the concept of algebra and must know its formulae very well. Students must know formulae in algebra like,
\[ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]
\[\Rightarrow {{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)\].
Here students should not make mistake like using the formula wrong for example if the students use \[ {{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a-b \right)\] this instead of \[ {{a}^{3}}+{{b}^{3}}={{\left( a+b \right)}^{3}}-3ab\left( a+b \right)\] our whole solution will be wrong as this formula is the major step in our solution. So, we must take care and not make mistakes of this kind.