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Chapter 7: Problem 75

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

Short Answer

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

Step by step solution

01

Factoring the Numerator and the Denominator

Start by factoring the polynomial in the numerator and the denominator. The expression will then become: \[\frac{(x - y)(x + 3y)}{(2x - y)(x + 3y)} \]
02

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:\[\frac{x - y}{2x - y} \]
03

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 \(\frac{x - y}{2x - y}\)

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x+6}-1$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{7}{x}+4$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{10}{x+3}-\frac{2}{-x-3}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6 x+5}{x-2}+\frac{4 x}{2-x}$$
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