Question

How do you solve for the given fraction into the simplest possible form, the given fraction is \[\dfrac{{{x^5} + {y^5}}}{{x + y}}\]?

Answer 1

Hint: The given question is of solving the fraction in terms of algebraic expression, here for such simplification you have to either expand the terms if possible or can for direct cancelling if terms are arranged like that, the final answer may be in fraction or no fraction is available, but it should be the simplest possible term.

Complete step by step solution:
The given question is \[\dfrac{{{x^5} + {y^5}}}{{x + y}}\]. Here we are solving this we cannot factorize the above expression only direct division is possible that means how we solve for the division of two numbers with the standard division method like that only we have to solve for the expression, only the thing is here we have to deal with the algebraic division instead of numerical division, on solving we get:
\[{x^5} + {y^5} \div x + y = {x^4} - {x^3}y + {x^2}{y^2} - x{y^3} + {y^4}\]

Here we have done the division in the same manner as we do for the number system, in such division you have to think about cancelling of the first term and if possible then two terms together, here both the terms cannot be cancelled together, so we move further cancelling the first term only and finally we achieve the zero remainder.

Note: When factorization is not possible for the algebraic expressions then you need to solve it directly, for which you have to be careful that without thinking the second term and the remainder for each step just think for the dividend which can cancel out completely the first term and move forward.