Question

How do you factor the expression \[{x^3}{y^3} + {z^3}\]?

Answer 1

Hint: Factorizing reduces the higher degree equation into its linear equation. In the above given question, we need to reduce the equation where the highest power is 3. It can be reduced by using the formula of sum of cubes which helps in reducing into the terms having power less than 3.

Complete step by step answer:
Given, the equation has terms with the highest power of 3. We can use the formula of the sum of cubes. The sum of two cubes can be factored into a product of a binomial times a trinomial.

The factoring formula for sum of cubes is as follows:
\[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\]
First, we need to group the first term as a whole cube because each term needs to be a perfect cube. Perfect cube means the term a should be the cube of another number. For example,8 is the cube of the number 2. Hence 8 will be a perfect square.
\[{x^3}{y^3} = {\left( {xy} \right)^3}\]
So, now here a becomes xy and b is z.
Now substituting the values of a and b we get,
\[{\left( {xy} \right)^3} + {z^3} = \left( {\left( {xy} \right) + z} \right)\left( {{{\left( {xy} \right)}^2} - \left( {xy} \right)z + {z^2}} \right)\]
Now squaring the individual term x and y by opening the parenthesis we get,
\[{\left( {xy} \right)^3} + {z^3} = \left( {xy + z} \right)\left( {{x^2}{y^2} - xyz + {z^2}} \right)\]
Therefore, we get the above reduced equation.

Note: It is always important while solving a third power polynomial to understand that the equation belong to which different factorizing scenarios like whether it is sum of two cubes or difference of two cubes and identifying a and b in such a way that the terms should be perfect cubes.