Question

How do you factor \[{{y}^{3}}+27?\]

Answer 1

Hint: To solve the equation or to factor the above given equation we have to first understand the above equation form.
The above equation can be written in the form \[{{a}^{3}}+{{b}^{3}}\] and we can use the direct formula.

Complete step by step solution:
Given equation is \[{{y}^{3}}+27?\]
We can write the above equation in the form at \[{{a}^{3}}+{{b}^{3}}.\]
Here \[a=y\] and \[b=3\]
\[{{y}^{3}}+{{\left( 3 \right)}^{3}}\]
Now, we know that the formula of \[{{a}^{3}}+{{b}^{3}}\] is
\[\Rightarrow {{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]
\[\Rightarrow {{y}^{3}}+{{3}^{3}}=\left( y+3 \right)\left( {{y}^{2}}-3y+{{3}^{2}} \right)\]
We know that the value at \[{{3}^{2}}\] is \[9\].
\[\therefore {{y}^{3}}+{{3}^{3}}=\left( y+3 \right)({{y}^{2}}-3y+9)\]
The \[{{y}^{3}}+27\] is factored as
\[\therefore {{y}^{3}}+27=\left( y+3 \right)({{y}^{2}}-3y+9)\]

Note: There are two formulas of cube one if for \[{{a}^{3}}+{{b}^{3}}\] which is used here and another one is \[{{a}^{3}}-{{b}^{3}}\] the formula of \[{{a}^{3}}-{{b}^{3}}\] is \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)\].
The difference between two formulas is only positive and negative signs so do not get confused between these two formulas.