Question

Find the following product.
 $ \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) $

Answer 1

Hint: We know that expansion of $ \left( {a - b} \right)\left( {c + d} \right) $ can be obtained by multiplying each term of first expression with each term of second expression. We will use this information to find the required product.

Complete step-by-step answer:
In this problem, we have two mathematical expressions $ {x^3} - {y^3} $ and $ {x^2} + {y^2} $ . We have to find the product of these two expressions. That is, we have to expand $ \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) $ . Now expansion of $ \left( {a - b} \right)\left( {c + d} \right) $ is obtained by multiplying each term of first expression $ \left( {a - b} \right) $ with each term of second expression $ \left( {c + d} \right) $ . That is,
 $ \left( {a - b} \right)\left( {c + d} \right) = a\left( {c + d} \right) - b\left( {c + d} \right) $
 $ \Rightarrow \left( {a - b} \right)\left( {c + d} \right) = ac + ad - bc - bd $
Let us expand $ \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) $ in the same manner. So, we can write
 $ \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) = {x^3}\left( {{x^2} + {y^2}} \right) - {y^3}\left( {{x^2} + {y^2}} \right) $
 $ \Rightarrow \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) = \left( {{x^3} \times {x^2}} \right) + \left( {{x^3} \times {y^2}} \right) - \left( {{y^3} \times {x^2}} \right) - \left( {{y^3} \times {y^2}} \right) \cdots \cdots \left( 1 \right) $
We know that $ {a^m} \times {a^n} = {a^{m + n}} $ . This is called the law of exponents. Use this law in the first and last bracket of RHS of the equation $ \left( 1 \right) $ . So, we can write
 $ \left( {{x^3} - {y^3}} \right) \times \left( {{x^2} + {y^2}} \right) = {x^5} + {x^3}{y^2} - {y^3}{x^2} - {y^5} $
Hence, we can say that the product of $ {x^3} - {y^3} $ and $ {x^2} + {y^2} $ is $ {x^5} + {x^3}{y^2} - {y^3}{x^2} - {y^5} $ .

Note: In this type of problems, we have to find the required product by multiplying each term of one expression with each term of another expression. In this problem, one mathematical expression is $ {x^3} - {y^3} $ . Remember that the factorization of $ {x^3} - {y^3} $ is given by $ {x^3} - {y^3} = \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) $ . Expansion of $ \left( {a + b} \right)\left( {c + d} \right) $ is given by $ ac + ad + bc + bd $ .