Problem 82 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \((y - 7)(x + 4)\).

Step 3: Factor the Denominator

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

Step 4: 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.