Problem 75 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+2 x y-3 y^{2}}{2 x^{2}+5 x y-3 y^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(\frac{x - y}{2x - y}\)

Step 1: Factoring the Numerator and the Denominator

Start by factoring the polynomial in the numerator and the denominator. The expression will then become: \[\frac{(x - y)(x + 3y)}{(2x - y)(x + 3y)} \]

Step 2: Cancelling out the common factors

Once the expressions have been factored, the next step is to cancel out the common factors, which is (x + 3y) in this case. The expression will then simplify to:\[\frac{x - y}{2x - y} \]

Step 3: Checking if further simplification is possible

The simplified form of the given rational expression is \(\frac{x - y}{2x - y} \). Since there are no common factors left in the numerator and the denominator, further simplification is not possible. Thus, the final simplified form of the given rational expression is \(\frac{x - y}{2x - y}\)