Question

How do you divide $\dfrac{{{x^5} - {x^5}}}{{x - y}}$

Answer 1

Hint: Assuming the second \[{x^5}\]should have been a\[{y^5}\]:
If we continue with the same given equation, the value becomes zero immediately in the numerator and gives the resultant value as zero. So, we assume second \[{x^5}\]should have been a\[{y^5}\]. Then we will be able to divide the expression to get its simplified form, in which Both the numerator and denominator are homogeneous polynomials in x and y; the numerator being of degree 6 and the denominator of degree 1. With the help of a general form, we derive an equation for the numerator of the given expression and simplify, which gives us the answer.

Complete step-by-step solution:
We are given that an expression $\dfrac{{{x^5} - {x^5}}}{{x - y}}$
Since, we know the formula for expressions like
\[
\Rightarrow {a^2} - {b^2} = (a + b)(a - b) \\
 \Rightarrow {a^3} - {b^3} = (a - b)({a^2} + ab + {b^2}) \\
\]
And we know a general form from which we can expand the numerator is
\[\Rightarrow {x^n} - {y^n} = (x - y)({x^{n - 1}} + {x^{n - 2}}y + ... + x{y^{n - 2}} + {y^{n - 1}})\]
From, this we can derive the formula for the numerator in the given expression, we get
\[\Rightarrow {x^5} - {y^5} = (x - y)({x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4})\]
Now, we will substitute this in the numerator now, which will be
$\Rightarrow \dfrac{{(x - y)({x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4})}}{{x - y}}$
Now, we will look for any common terms which can be cancelled and then we can further simplify it
And after we have cancelled the equations, we will get
$\Rightarrow ({x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4})$
This is the final expression we get on dividing the given expression in the question.

Note: We have to be careful, while finding out the formula from the general form, as while substituting in the form, we might make mistakes in the power place and we have to re-check again for mistakes after substitution.