Chapter 5.4, Problem 35WE

To determine

To simplify the given expression.

Answer to Problem 35WE

  $\frac{(x^3-y^3)}{(x-y)(x^2+y^2)}$

Explanation of Solution

Given:

Expression: $\frac{x^4+x^{3}y-x{y}^3-y^4}{x^4-y^4}$

Concept used:

To simplify the given expression:

1. Take out common terms in the given expression.
2. Cancel out the terms that are in common in numerator and the denominator.
3. Find if any algebraic identities can simplify the given expression.

Calculation:

In the given expression find out the common term in the numerator and denominator.

  $\Rightarrow \frac{x^4+x^{3}y-x{y}^3-y^4}{x^4-y^4}$

Consider the numerator and the denominator separately and factorize.

Consider the numerator and factorize.

  $\begin{array}{l}\Rightarrow x^4+x^{3}y-x{y}^3-y^4\\ \Rightarrow (x^4+x^{3}y)-(x{y}^3+y^4)\\ \Rightarrow x^3(x+y)-y^3(x+y)\\ \Rightarrow (x^3-y^3)(x+y)\end{array}$

Consider the denominator and factorize.

  $\begin{array}{l}\Rightarrow x^4-y^4\\ \Rightarrow (x^2-y^2)(x^2+y^2)\end{array}$

Substituting the factorized form of the numerator and the denominator in the given expression,

  $\begin{array}{l}\Rightarrow \frac{(x^3-y^3)(x+y)}{(x^2-y^2)(x^2+y^2)}\\ \Rightarrow \frac{(x^3-y^3)(x+y)}{(x-y)(x+y)(x^2+y^2)}\\ \Rightarrow \frac{(x^3-y^3)\cancel{(x+y)}}{(x-y)\cancel{(x+y)}(x^2+y^2)}\\ \Rightarrow \frac{(x^3-y^3)}{(x-y)(x^2+y^2)}\end{array}$

Conclusion:

Therefore, the simplified expression is $\frac{(x^3-y^3)}{(x-y)(x^2+y^2)}.$