Problem 63 - Elementary and Intermediate Algebra

Question

Simplify. See Example \(8 .\) $$ \frac{2-a}{a^{2}-a-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{1}{a+1}\).

Step 1: Identify the Denominator

The denominator of the fraction is the expression \(a^{2} - a - 2\).

Step 2: Factor the Quadratic Expression

The quadratic expression \(a^{2} - a - 2\) factors into two binomials. We look for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\), so \(a^{2} - a - 2 = (a-2)(a+1)\).

Step 3: Rewrite the Fraction

Substitute the factored form of the denominator back into the fraction. This gives \[ \frac{2-a}{(a-2)(a+1)} \]

Step 4: Simplify the Fraction

Notice that the numerator \(2-a\) can be rewritten as \(-1(a-2)\). This is because \(2-a = -(a-2)\). Thus, the fraction becomes \[-\frac{a-2}{(a-2)(a+1)}\]

Step 5: Cancel Common Factors

The expression \(a-2\) appears in both the numerator and the denominator, so they can be cancelled out, resulting in \[ -\frac{1}{a+1} \]

Step 6: Present the Simplified Expression

The fraction simplifies to \[ -\frac{1}{a+1} \]

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is an essential skill in algebra. The goal is to make expressions as concise and clear as possible, which often involves reducing fractions and combining like terms. When simplifying fractions, we aim to write them in their simplest form. This usually involves factoring and canceling out any common factors between the numerator and the denominator.

In the context of our original exercise, we began with the expression \( \frac{2-a}{a^{2}-a-2} \). The initial step was to factor the denominator, recognizing it as a quadratic that could be rewritten as a product of two binomials. Once rewritten as \((a-2)(a+1)\), it allowed us to see common factors more clearly, which is crucial for the simplification process.

By rewriting the numerator \(2-a\) as \(-1(a-2)\), we located a common factor that could be canceled, ultimately simplifying the expression.

Simplifying saves time and reduces complexity, making further calculations or comparisons easier.

Factoring Quadratics

Factoring quadratics is a method used to express a quadratic equation in a simpler form by finding its binomial factors. This process is especially useful in simplifying fractions where a quadratic is in the denominator.

Let's examine the quadratic expression \(a^2 - a - 2\). To factor it, we needed numbers that multiply to the constant term \(-2\) and add up to the linear coefficient \(-1\).

• Here, the numbers were \(-2\) and \(+1\).
• This allowed us to break the quadratic into \((a-2)(a+1)\).

Factoring transforms the expression into an easier-to-handle format, revealing factors common with the numerator.Using this skill simplifies complex algebraic fractions and is a core concept in algebra that supports solving equations and inequalities and graphing parabolas on the coordinate plane.

Canceling Common Factors

Canceling common factors is a significant simplification strategy in algebra. It involves eliminating identical elements present in both the numerator and the denominator of a fraction, thereby reducing it to its simplest form.

Take our fraction from the exercise: \[-\frac{a-2}{(a-2)(a+1)}\]. The factor \(a-2\) was in both the top and the bottom. By canceling these out, we simplified the fraction to \(-\frac{1}{a+1}\).

• Canceling is only applicable when the same factor appears in the numerator and denominator.
• This process should not change the value of the expression, aside from its complexity and appearance.

This technique is powerful for reducing complex rational expressions, enabling easier calculations later on or solving more complicated algebraic equations with higher efficiency.