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Chapter 7: Problem 82

Simplify each rational expression. $$\frac{x y+4 y-7 x-28}{x^{2}+11 x+28}$$

Short Answer

Expert verified
The simplified form of the given rational expression is \( \frac{y - 7}{x + 7} \).

Step by step solution

01

Analyze the Rational Expression

The given expression is \(\frac{x y+4 y-7 x-28}{x^{2}+11 x+28}\). We have a polynomial of two terms in the numerator and a quadratic polynomial in the denominator.
02

Factor the Numerator

The numerator \(x y+4 y-7 x-28\) can be factored by grouping. If we rearrange the terms as \((x y - 7 x) + (4 y - 28)\), it can be rewritten as \(x(y - 7) + 4(y - 7)\). Therefore, the factored form of the numerator is \((y - 7)(x + 4)\).
03

Factor the Denominator

The denominator \(x^{2}+11 x + 28\) is a quadratic polynomial and can be factored as \((x + 4)(x + 7)\).
04

Simplify the Rational Expression

After factoring both the numerator and the denominator, the rational expression becomes \(\frac{(y - 7)(x + 4)}{(x + 4)(x + 7)}\). Cancel out the common factor, which is \(x + 4\), to simplify the rational expression.

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

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