91Ó°ÊÓ

Factoring quadratics is a fundamental skill in algebra that involves breaking down a quadratic expression, generally in the form of \( ax^2 + bx + c \), into simpler, multiplicative components. In this particular exercise, we start with the quadratic \( x^2 + 3x + 2 \) in the denominator.

Our goal is to transform this expression into a product of two binomials. The technique used here involves finding two numbers that multiply to the constant term (\( c = 2 \)) while adding to the linear coefficient (\( b = 3 \)).

With this in mind:

• The two numbers that satisfy these conditions are 1 and 2, since \( 1 \times 2 = 2 \) and \( 1 + 2 = 3 \).

We can express \( x^2 + 3x + 2 \) as \( (x+1)(x+2) \).

This decomposition or factoring simplifies later stages of manipulation, as seen in our rational expression.

Simplifying rational expressions involves reducing them to their most basic form. By simplifying, we make these expressions easier to work with and understand.

In our exercise, after factoring the quadratic in the denominator to \( (x+1)(x+2) \), the rational expression becomes \( \frac{x+2}{(x+1)(x+2)} \).

Here, the \( x+2 \) component in the numerator and denominator cancel each other out because they are identical. It's important to realize that you can only cancel a common factor from both parts if it does not make the denominator zero.

• After cancellation, the expression simplifies to \( \frac{1}{x+1} \).

It's always essential to check for restrictions on \( x \) in such operations, which leads us to the next critical concept.

In mathematics, division by zero is undefined. It's crucial to ensure that the operations leading to division never result in a zero in the denominator.

When simplifying rational expressions, like in our exercise, it's necessary to identify the values of \( x \) that could potentially make the denominator zero. After factoring the quadratic from the original expression, the factored denominator is \( (x+1)(x+2) \).

From this form, you can see:

• If \( x = -1 \), the expression \( (x+1) = 0 \).
• If \( x = -2 \), the expression \( (x+2) = 0 \).

Thus, \( x \) cannot equal \(-1\) or \(-2\), since these values would cause division by zero. The simplified expression \( \frac{1}{x+1} \) specifically highlights that \( x eq -1 \), maintaining the integrity of the expression and avoiding undefined behavior.