Problem 57 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{x^{2}+7 x-18}{12-4 x-x^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{x + 9}{x + 6}\).

Step 1: Write the problem in a standard form

Given the algebraic expression, \( \frac{x^{2}+7x-18}{12-4x-x^{2}} \), notice that the numerator \(x^2 + 7x - 18\) and the denominator \(12 - 4x - x^2\) are both quadratic polynomials.

Step 2: Factor the numerator

To factor the numerator \(x^2 + 7x - 18\), we look for two numbers that multiply to \(-18\) (the constant term) and add to \(7\) (the coefficient of the \(x\) term). Those numbers are \(9\) and \(-2\). Therefore, the numerator factors to \((x + 9)(x - 2)\).

Step 3: Factor the denominator

To factor the denominator \(12 - 4x - x^2\), first rewrite it in standard form: \(-x^2 - 4x + 12\). It's often easier to factor if the leading coefficient is positive, so factor out a \(-1\): \(-(x^2 + 4x - 12)\). Now, factor \(x^2 + 4x - 12\) by finding two numbers that multiply to \(-12\) and add to \(4\). Those numbers are \(6\) and \(-2\), so the expression factors to \(-((x + 6)(x - 2))\).

Step 4: Simplify the expression

Now that we have factored both the numerator and the denominator, the expression is: \[\frac{(x + 9)(x - 2)}{-(x + 6)(x - 2)}\]. The \((x - 2)\) term appears in both the numerator and the denominator and can be canceled out. Be mindful of the negative sign in front of the denominator. After cancellation, the expression becomes \(-\frac{x + 9}{x + 6}\).

Key Concepts

Factoring Quadratic Polynomials

When simplifying algebraic fractions, a key step is factoring quadratic polynomials. A quadratic polynomial is typically expressed in the form \(ax^2 + bx + c\). Factoring these polynomials involves expressing them as a product of two binomials.

Let's consider the exercise at hand. The numerator is \(x^2 + 7x - 18\). To factor it, we look for two numbers that multiply to \(-18\) (the constant term) and add to \(7\) (the coefficient of the middle term, \(x\)). These numbers are \(9\) and \(-2\). Thus, the expression can be factored into \((x + 9)(x - 2)\).

In the denominator, we have \(12 - 4x - x^2\), which we first write in a more standard form as \(-x^2 - 4x + 12\). Factoring out the negative sign gives \(-(x^2 + 4x - 12)\). Again, we need two numbers that multiply to \(-12\) and add to \(4\). These numbers are \(6\) and \(-2\), so the factorization becomes \((x + 6)(x - 2)\). Applying the negative factor to this, we find the complete factorization of the denominator is \(-(x + 6)(x - 2)\).

Remember that factoring polynomials is a critical skill in algebra, helpful not just for simplifying expressions but also for solving equations and understanding functions.

Canceling Common Factors

Once both the numerator and the denominator of the algebraic fraction are factored, the next step is to cancel common factors. This process simplifies the fraction by removing identical expressions that appear in both parts.

For our example, after factoring, we have:

• Numerator: \((x + 9)(x - 2)\)
• Denominator: \(-(x + 6)(x - 2)\)

Notice the common factor \((x - 2)\) in both the numerator and the denominator. Cancelling this common factor reduces the fraction to \(-\frac{x + 9}{x + 6}\), where the negative sign is a result of the negative factored out in the denominator.

Canceling is straightforward:

• Identify terms that are exactly the same in both the numerator and the denominator.
• Remove one occurrence of the common factor from each.

It's essential to remember that cancellation works because any non-zero expression divided by itself is \(1\), simplifying the fraction without changing its value. Just be cautious with signs—keep track of any negative signs when they are involved in the cancellation process.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations like addition, subtraction, multiplication, and division. These expressions can be simplified using algebraic rules and techniques.

In our exercise, the expression \(\frac{x^2+7x-18}{12-4x-x^2}\) is an algebraic fraction, which includes both a numerator and a denominator that are algebraic expressions. Simplifying such fractions involves several steps, including rewriting the terms in standard form, factoring, and reducing by canceling common factors.

Understanding how to manipulate these expressions is crucial in algebra, as it enables:

• Solving equations and inequalities by simplifying complex expressions.
• Understanding relationships between variables.
• Breaking down polynomial functions into simpler components for analysis.

When dealing with algebraic expressions, always keep the order of operations in mind: first handle factors and powers, then perform multiplications and divisions from left to right, followed by additions and subtractions. Practicing these techniques strengthens problem-solving skills and builds a solid foundation in mathematics.