Question

How do you factor \[125{{x}^{3}}+343{{y}^{3}}\]?

Answer 1

Hint: The expressions of the form \[{{a}^{3}}+{{b}^{3}}\] are called the addition of cubes. To solve the given question, we need to know the algebraic expansion of \[{{a}^{3}}+{{b}^{3}}\]. This expression is expanded as \[{{a}^{3}}+{{b}^{3}}=(a+b)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]. For the given expression we have \[a=5x\And b=7y\]. By substituting these values in the expansion, we can factorize the expression.

Complete step by step answer:
We are asked to factorize the expression \[125{{x}^{3}}+343{{y}^{3}}\].
We know that 125 is the cube of 5, and 343 is cube of 7. Using this in the above expression, it can be written as \[{{5}^{3}}{{x}^{3}}+{{7}^{3}}{{y}^{3}}\]. Using the algebraic property \[{{a}^{n}}{{b}^{n}}={{\left( ab \right)}^{n}}\], we can simplify this expression as
\[\Rightarrow {{\left( 5x \right)}^{3}}+{{\left( 7y \right)}^{3}}\]
This expression is of the form of addition of cubes, we know that the expansion of the form \[{{a}^{3}}+{{b}^{3}}=(a+b)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\]. Here, we have \[a=5x\And b=7y\]. substituting the values of a and b for this question in the expansion of addition of cubes, we get
\[{{\left( 5x \right)}^{3}}+{{\left( 7y \right)}^{3}}=\left( 5x+7y \right)\left( {{\left( 5x \right)}^{2}}-(5x)(7y)+{{\left( 7y \right)}^{2}} \right)\]
We know that the square of 5 is 25, substituting this value above and, simplifying the above expression, we get
\[{{\left( 5x \right)}^{3}}+{{\left( 7y \right)}^{3}}=\left( 5x+7y \right)\left( 25{{x}^{2}}-35xy+49{{y}^{2}} \right)\]

Thus, the factored form of the given expression is
\[{{\left( 5x \right)}^{3}}+{{\left( 7y \right)}^{3}}=\left( 5x+7y \right)\left( 25{{x}^{2}}-35xy+49{{y}^{2}} \right)\].

Note: We should know the expansion of the following expressions, the expansions for these expressions are as follows, the difference of square \[{{a}^{2}}-{{b}^{2}}\] is expanded as \[\left( a+b \right)\left( a-b \right)\]. The difference of cubes is expanded as \[{{a}^{3}}-{{b}^{3}}=(a-b)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]. To use these expansions, we first have to find the value of a and b, then substitute it in the expansion.
The expansions of the form \[{{a}^{2n}}+{{b}^{2n}}\], here n is a positive integer. These can not be further factorized.