Problem 55 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{6-x-2 x^{2}}{12+7 x-10 x^{2}}$$

Step-by-Step Solution

Answer

\(\frac{x + 2}{5x + 4}\)

Step 1: Identify the GCD

First, recognize that the simplified form of a fraction involves dividing both the numerator and the denominator by a common factor. The greatest common divisor (GCD) must be identified or factored out from both the numerator and the denominator if possible.

Step 2: Factor the Numerator

For the expression in the numerator, which is \(-2x^2 - x + 6\), try factoring by rewriting it in simpler terms. Rearrange terms as \(-2x^2 - x + 6 = -(2x^2 + x - 6)\). Now factor using the AC method or trial and error to get \(-(2x - 3)(x + 2)\).

Step 3: Factor the Denominator

Similarly, factor the denominator. Consider the quadratic expression \(-10x^2 + 7x + 12\). We rearrange it as \(- (10x^2 - 7x - 12)\) and factor it. Use the AC method to get \( (2x - 3)(5x + 4)\), so the original expression is \(-(2x - 3)(5x + 4)\).

Step 4: Cancel the Common Factors

Now that we have factored expressions in numerator and denominator, compare. Both \(-(2x - 3)(x + 2)\) and \(-(2x - 3)(5x + 4)\) have a common factor of \( -(2x - 3)\). Cancel this common factor.

Step 5: Write the Simplified Fraction

After canceling the \(-(2x - 3)\) from both the numerator and the denominator, you are left with the simplified form:\[ \frac{x + 2}{5x + 4} \]

Key Concepts

Factoring

Factoring is the process of breaking down an expression into a product of simpler factors. It is a crucial step in simplifying algebraic fractions. When working with quadratic expressions, such as the one in the numerator \( -2x^2 - x + 6 \), factoring helps express it in a more manageable form.

In this instance, rearrange the expression to make it easier to factor: \( - (2x^2 + x - 6) \). Now, using techniques like the AC method, where you look for two numbers that multiply to the product of the first and last coefficients, you find that the expression factors to \( - (2x - 3)(x + 2) \).

Understanding how to rearrange and factor expressions like these is essential for simplifying algebraic fractions. It often involves looking for patterns, grouping terms, and using methods such as factoring by grouping or using special products like difference of squares.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest factor that divides two or more numbers. When simplifying algebraic fractions, finding the GCD of the numerator and the denominator is a key step.

• First, recognize common factors in both parts of the fraction.
• Identify the GCD to simplify the fraction.

In the given problem, the common factor in both the numerator and denominator is \( -(2x - 3) \). This commonality means we can divide both the top and bottom by \( -(2x - 3) \) without changing the value of the fraction.

Understanding the role of the GCD is crucial because it allows us to reduce complex expressions into simpler forms, ensuring fractions are as simple as possible.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves canceling out any common factors between the numerator and denominator. In our exercise, we start with the factored forms of both numerator and denominator:

Numerator is \( - (2x - 3)(x + 2) \), while the denominator is \( - (2x - 3)(5x + 4) \). Identifying the \(-(2x - 3)\) common factor allows us to cancel it, which results in the simplified fraction \( \frac{x + 2}{5x + 4} \).

When simplifying, it’s critical to ensure that only non-zero common factors are canceled. Remember, the act of simplifying should preserve the value of the original expression, making it easier to work with and interpret.