Question

What are the factors of \[{{a}^{3}}-{{b}^{3}}\]?

Answer 1

Hint: We solve this problem by using the factorization method from the formula of the difference of cube of two numbers. The formula of the difference of cube of two numbers is given as
\[{{x}^{3}}-{{y}^{3}}={{\left( x-y \right)}^{3}}+3xy\left( x-y \right)\]
Then we take the common terms out so as to reduce it into a product of two factors.
We also use the formula of the square of the difference of two numbers that are
\[{{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}\]
We reduce the given polynomial as a product of two other polynomials which can’t be reduced any more to conclude them as the factors of the given polynomial.

Complete step-by-step solution
We are asked to find the factors of \[{{a}^{3}}-{{b}^{3}}\]
Let us assume that the given polynomial as
\[\Rightarrow P={{a}^{3}}-{{b}^{3}}\]
We know that the formula of difference of cube of two numbers is given as
\[{{x}^{3}}-{{y}^{3}}={{\left( x-y \right)}^{3}}+3xy\left( x-y \right)\]
By using this formula to given polynomial we get
\[\Rightarrow P={{\left( a-b \right)}^{3}}+3ab\left( a-b \right)\]
Now, by taking the common term out we get
\[\Rightarrow P=\left( a-b \right)\left( {{\left( a-b \right)}^{2}}+3ab \right)\]
We know that the standard formula of square of difference of two numbers that is
\[{{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}\]
By using this formula to above equation we get
\[\begin{align}
  & \Rightarrow P=\left( a-b \right)\left( {{a}^{2}}-2ab+{{b}^{2}}+3ab \right) \\
 & \Rightarrow P=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right) \\
\end{align}\]
Here, we can see that the above product cannot be reduced to product of some other polynomials.
So, we can see that given polynomial can be written as
\[\therefore {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]
Therefore, the factors of given polynomial are \[\left( a-b \right)\] and \[\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]

Note: We have a shortcut for the above formula.
We are asked to find the factors of \[{{a}^{3}}-{{b}^{3}}\]
The formula for difference of cube of two numbers is given as
\[{{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)\]
By using this formula directly we can write the given polynomial as
\[\therefore {{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]
Therefore, the factors of given polynomial are \[\left( a-b \right)\] and \[\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]