Problem 23 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x^{2}+x-12}{x^{2}-6 x+9}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{x^{2}+x-12}{x^{2}-6x+9}\) is \(x+4\).

Step 1: Factor the Numerator

Begin by factoring the numerator \(x^{2}+x-12\). This can be done by identifying two numbers that multiply together to be -12 and add together to be 1. The numbers 4 and -3 satisfy these conditions, so the numerator factors as \(x^2 + x - 12 = (x-3)(x+4)\)

Step 2: Factor the Denominator

Next, factor the denominator \(x^{2}-6x+9\). This is accomplished by identifying two numbers that multiply together to be 9 and add together to be -6. The numbers -3 and -3 satisfy these conditions, hence the denominator factors as \(x^2 -6x + 9 = (x-3)(x-3) = (x-3)^2\)

Step 3: Cancel out Common Factors

After factoring the numerator and denominator, (x-3) appears in both. Cancel out this common factor to simplify the expression. The expression, thus, simplifies to \(\frac{x-3}{x-3} * \frac{x+4}{1} = x+4\).

Key Concepts

Factoring

Factoring is a crucial step in simplifying rational expressions, where we break down polynomials into products of simpler factors. This process allows us to identify common factors that can be canceled to simplify the expression. For example, the expression given in the exercise, \[ x^2 + x - 12 \], can be factored by finding two numbers that multiply to -12 and add to 1. These numbers are 4 and -3, leading us to factor the numerator as \( (x+4)(x-3) \).
To factor polynomials effectively:

• Examine the constant term to find possible factor pairs.
• Choose a pair of factors that add up to the linear coefficient.
• Rewrite the polynomial as a product of two binomials.

This skill helps in simplifying rational expressions by making "invisible" common factors visible and thus removable.

Simplifying Rational Expressions

Simplifying rational expressions means reducing them to their simplest form. This involves factoring both the numerator and the denominator to cancel out common factors. To simplify the rational expression, follow these steps:

• Factor both the numerator and the denominator completely.
• Identify and cancel common factors between the numerator and the denominator.
• Write the remaining factors as the simplified expression.

For example, in the expression \( \frac{x^2 + x - 12}{x^2 - 6x + 9} \), after factoring, we get \( \frac{(x+4)(x-3)}{(x-3)^2} \). The common factor \( (x-3) \) appears in both the numerator and the denominator, so it can be canceled. This simplifies the expression to \( x+4 \), which is now much easier to work with. Simplifying these expressions is vital to solving algebra problems more efficiently.

Common Factors

Common factors play a key role in simplifying rational expressions. A common factor is a factor that appears in both the numerator and the denominator of a rational expression. By identifying and canceling these common factors, we reduce the expression to its simplest form.
Identifying common factors involves comparing the factorized forms of polynomials. In our example, after factoring \( x^2 + x - 12 \) into \( (x+4)(x-3) \) and \( x^2 - 6x + 9 \) into \( (x-3)^2 \), the common factor \( (x-3) \) appears in both. By canceling this factor, we are left with \( x+4 \), which is the simplest form of the expression without any more common factors to cancel.
Using common factors to simplify rational expressions makes mathematical calculations more manageable and results more interpretable.