Problem 20 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{4\left(x^{2}-1\right)}{12(x+2)(x-1)} $$

Step-by-Step Solution

Answer

\(\frac{x+1}{3(x+2)}\)

Step 1: Factor the Quadratic Expression

The expression in the numerator is \(4(x^2 - 1)\). We notice that \(x^2 - 1\) is a difference of squares, which can be written as \((x - 1)(x + 1)\). Therefore, the numerator can be factored as \(4(x - 1)(x + 1)\).

Step 2: Rewrite the Expression

Now the entire rational expression becomes \(\frac{4(x-1)(x+1)}{12(x+2)(x-1)}\).

Step 3: Cancel Common Factors

In the rational expression \(\frac{4(x-1)(x+1)}{12(x+2)(x-1)}\), both the numerator and the denominator have a common factor of \((x - 1)\). We can cancel \((x - 1)\) from both the numerator and the denominator.

Step 4: Simplify the Remaining Expression

After canceling \((x - 1)\), the expression becomes \(\frac{4(x + 1)}{12(x + 2)}\). We can simplify this further by reducing the constant factors. The greatest common divisor of 4 and 12 is 4, so divide both by 4 to get \(\frac{(x + 1)}{3(x + 2)}\).

Step 5: Final Result

The simplified rational expression is \(\frac{x+1}{3(x+2)}\). This is the final answer as no further simplification can be done.

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental concept in algebra that helps break down complex expressions. In the given problem, we begin by recognizing the form of the quadratic expression. The term \(x^2 - 1\) in the numerator is a perfect candidate for factoring. This particular form is known as a difference of squares.
This means it can be expressed as two binomials: \((x - 1)(x + 1)\).
To factor a quadratic, search for patterns:

• Look for a squared term and its constant difference.
• Use the identity \(a^2 - b^2 = (a - b)(a + b)\).

This step is crucial for simplifying rational expressions, as it makes potential cancellation of terms much easier.

Difference of Squares

The difference of squares is an algebraic pattern that appears frequently in mathematics. Here, \(x^2 - 1\) is identified as a difference of squares. The formula to remember is \(a^2 - b^2 = (a - b)(a + b)\).
Applying this to our example, where \(a = x\) and \(b = 1\), we factor \(x^2 - 1\) into \((x - 1)(x + 1)\).
Recognizing such patterns:

• Makes simplifying expressions much more straightforward.
• Enables you to see opportunities for canceling common factors later in the problem.

Be on the lookout for squares and differences in expressions to utilize this powerful factoring technique.

Canceling Common Factors

One of the critical steps in simplifying rational expressions is canceling out common factors found in both the numerator and the denominator. After factoring the expression, we identify that both the numerator and the denominator include the factor \((x - 1)\).

Here's how you can perform cancelation effectively:

• Ensure that the factor appears in both parts of the fraction.
• "Cancel out" these factors by essentially dividing them out of the fraction, simplifying the expression.

This step not only streamlines the expression significantly but also reveals the true simplified form. It is important, however, to remember that you can only cancel factors, not terms.

Reducing Expressions

Reducing expressions involves simplifying them to their most basic form, making them easier to understand and work with. After canceling common factors from both the numerator and the denominator, further simplify by reducing numerical coefficients.
In the expression \(\frac{4(x + 1)}{12(x + 2)}\), the numbers 4 and 12 have a greatest common divisor of 4.
Thus, dividing both the numerator and denominator by 4, we get \(\frac{x + 1}{3(x + 2)}\).
Steps to follow for reducing expressions:

• Divide numerical coefficients by their greatest common divisor.
• Ensure all like terms and factors are as simplified as possible.

Reduction helps in achieving the simplest form of the expression, ensuring there's no further simplification possible.