Question

Which of the following is the factor of \[{\left( {x + y} \right)^3} - \left( {{x^3} + {y^3}} \right)\] ?
A. \[{x^2} + {y^2} + 2xy\]
B. \[{x^2} + {y^2} - xy\]
C. \[x{y^2}\]
D. \[3xy\]

Answer 1

Hint: First use the cubic algebraic formula to expand the given algebraic expression and then find their factor. We know the cube algebraic formula is given by the formula \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\] so by using this formula we will expand \[{\left( {x + y} \right)^3}\] . After expanding the algebraic equation then reduce the expression by subtracting it by other terms in the expression. After obtaining the factors we will compare those factors with the given factors in the options and then we will verify the correct option.

Complete step-by-step answer:
Given the algebraic expression \[{\left( {x + y} \right)^3} - \left( {{x^3} + {y^3}} \right)\]
Now we can see the first term of the given expression is in cubic form and it will be difficult to find the factor of expression without solving the cube so as we know the cube algebraic formula is given by the formula \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\] , hence we can write the given algebraic expression as
 \[
   = {\left( {x + y} \right)^3} - \left( {{x^3} + {y^3}} \right) \\
   = {x^3} + {y^3} + 3xy\left( {x + y} \right) - \left( {{x^3} + {y^3}} \right) \\
 \]
We can also write the above expression as
 \[ = {x^3} + {y^3} + 3xy\left( {x + y} \right) - {x^3} - {y^3}\]
By solving the expression, we can write
 \[ = 3xy\left( {x + y} \right)\]
Hence we can say the factors of the given expression are \[\left( {x + y} \right)\] and \[3xy\] , now from the given options in the question we can say option D is correct since one of the factors of the given expression is \[3xy\] .
So, the correct answer is “Option D”.

Note: It is interesting to note that whenever an expression is split into their factors, if we multiply those factors together we will get the expression whose factors were found, this method is used to verify whether the factor for the expression is correct or not.