(9x^4+3x^3y-5x^2y^2+xy^3)/(3x^3+2x^2y-xy^2)

4 min read Jun 16, 2024
(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.

  • 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.