Simplifying Rational Expressions: A StepbyStep Guide
This article will guide you through the process of simplifying the rational expression:
(9x^4 + 3x^3y  5x^2y^2 + xy^3) / (3x^3 + 2x^2y  xy^2)
Understanding the Basics
A rational expression is a fraction where both the numerator and denominator are polynomials. Simplifying these expressions involves finding common factors and canceling them out.
Step 1: Factor the Numerator and Denominator
To start, we need to factor both the numerator and the denominator.

Numerator:
 We can factor out a common factor of x from each term:
 x (9x^3 + 3x^2y  5xy^2 + y^3)

Denominator:
 We can also factor out a common factor of x from each term:
 x (3x^2 + 2xy  y^2)
Step 2: Identify and Cancel Common Factors
Now, we have:
[x (9x^3 + 3x^2y  5xy^2 + y^3)] / [x (3x^2 + 2xy  y^2)]
Notice that both the numerator and denominator have a common factor of x. We can cancel this out, leaving us with:
(9x^3 + 3x^2y  5xy^2 + y^3) / (3x^2 + 2xy  y^2)
Step 3: Factor the Remaining Expressions (if possible)
The next step is to try and factor the remaining expressions. The numerator is a cubic polynomial, and the denominator is a quadratic polynomial. Factoring these polynomials can be a bit more complex, but in this case, we can use the following strategies:

Numerator:
 Notice that the numerator can be factored by grouping.
 Group the first two terms and the last two terms: (9x^3 + 3x^2y) + (5xy^2 + y^3)
 Factor out the common factors from each group: 3x^2(3x + y)  y^2(5x  y)
 Factor out the common factor (3x + y): (3x + y)(3x^2  y^2)

Denominator:
 The denominator is a quadratic expression that can be factored into two binomials.
 After some experimentation, we find: (3x  y)(x + y)
Step 4: Simplify
Now, our expression looks like this:
[(3x + y)(3x^2  y^2)] / [(3x  y)(x + y)]
We can further simplify by recognizing that (3x^2  y^2) is a difference of squares and can be factored as (3x + y)(3x  y).
This gives us:
[(3x + y)(3x + y)(3x  y)] / [(3x  y)(x + y)]
Now we can cancel out the common factor (3x  y):
(3x + y)(3x + y) / (x + y)
Final Answer
The simplified form of the original expression is:
(3x + y)^2 / (x + y)
This represents the most simplified form of the rational expression.