%2F(3x%5E3%2B2x%5E2y-xy%5E2).jpeg "(9x^4+3x^3y-5x^2y^2+xy^3)/(3x^3+2x^2y-xy^2)")
Simplifying Rational Expressions: A Step-by-Step 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.
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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.