Problem 81 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{x y+2 y+3 x+6}{x^{2}+5 x+6}$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(\frac{y+3}{x+3}\), where \(x \neq -2\).

Step 1: Factorization of the numerator

Factorising the expression \(x y+2 y+3 x+6\), we can express this as \((x+2) y+3 (x+2)\), which can be further expressed as \((x+2)(y+3)\) upon taking \((x+2)\) as common.

Step 2: Factorization of the denominator

\(x^{2}+5 x+6\) is a quadratic equation. It can be factorised using the formula: \(ax^{2} + bx + c = a * (x - x_{1}) * (x - x_{2})\) where \(x_{1}\) and \(x_{2}\) are the roots of the quadratic equation. Here, \(x_{1} = -2\) and \(x_{2} = -3\), since the roots are the solutions to \(x^{2} + 5x + 6 = 0\). The factorization of the denominator then becomes \((x+2)(x+3)\).

Step 3: Simplifying the expression

Now we rewrite the whole expression as \(\frac{(x+2)(y+3)}{(x+2)(x+3)}\). Because the factor \((x+2)\) appears both in the numerator and the denominator, they can be reduced. The simplified expression becomes \(\frac{y+3}{x+3}\) provided \(x \neq -2\) to avoid zero in the denominator.