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

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Jul 03, 2024

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Simplifying the Rational Expression: (9x^4+3x^3y-5x^2y^2+xy^3)/(3x^3+2x^2y-xy^2)

Rational expressions are an essential part of algebra, and simplifying them is crucial in solving various mathematical problems. In this article, we will explore how to simplify the rational expression:

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

Step 1: Factorize the Numerator

To simplify the rational expression, we need to factorize the numerator. Let's break it down:

9x^4 + 3x^3y - 5x^2y^2 + xy^3

= (3x^3)(3x) + (3x^3)(y) - (x^2y^2)(5x) + (x^2y^2)(y)

= 3x^3(3x + y) - x^2y^2(5x - y)

Step 2: Factorize the Denominator

Now, let's factorize the denominator:

3x^3 + 2x^2y - xy^2

= x^2(3x) + x^2(2y) - xy(xy)

= x^2(3x + 2y) - xy(xy)

= x^2(3x + 2y) - x(y^2)

Step 3: Simplify the Rational Expression

Now that we have factorized both the numerator and denominator, we can simplify the rational expression:

(3x^3(3x + y) - x^2y^2(5x - y)) / (x^2(3x + 2y) - x(y^2))

= ((3x + y)(3x^3 - x^2y^2)) / (x(x(3x + 2y) - y^2))

= (3x + y) / (x - y)

Conclusion

After simplifying the rational expression, we get:

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

This simplified form is much easier to work with and can be used to solve various algebraic problems.