Problem 76 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+3 x y-10 y^{2}}{3 x^{2}-7 x y+2 y^{2}}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x+5y}{3x - y}\).

Step 1: Factorizing the Numerator

Start by trying to factorize the numerator, which resembles a quadratic expression of the form \(ax^2 + bx + c\). Subsequently, this equation turns into \((x+5y)(x-2y)\) through factorization.

Step 2: Factorizing the Denominator

Similarly, factorize the denominator, which also resembles a quadratic equation. The denominator turns into \((3x - 2y)(x - y)\) through factorization.

Step 3: Simplify the Rational Expression

Now having both the numerator and the denominator factorized, the goal is to simplify the complex fraction. Look for the common factors between the numerator and the denominator. The common factor is \((x-2y)\) which can be cancelled out to obtain the simplified rational expression.

Step 4: Final Simplification

After reducing common factors, we have \(\frac{x+5y}{3x - y}\) as the simplified rational expression.