Problem 50 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x^{2}+x}{x^{2}-4}+\frac{x^{2}-1}{x^{2}+5 x+6}$$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \((3x+2)/((x-2)(x+2)(x+3))\), where \(x \neq 2, -2, -3\)

Step 1: Factorize Quadratic Expressions

Firstly, it's necessary to factorize the quadratic expressions in the denominator of both fractions: \(x^{2}-4\) can be factored as \((x-2)(x+2)\), and \(x^{2}+5x+6\) as \((x+2)(x+3)\)

Step 2: Finding the Common Denominator

The next thing to do is to find a common denominator for these two fractions. Looking at the factorized forms of the two denominators, the common denominator should be \((x-2)(x+2)(x+3)\)

Step 3: Making Equivalent Fractions with the Common Denominator

Now we need to make each fraction have the common denominator. For the first fraction, multiply both the numerator and the denominator by \((x+3)\), and for the second fraction, multiply both the numerator and the denominator by \((x-2)\)

Step 4: Performing the Addition

Now that the two fractions have the same denominator, add the numerators together. After simplification, divide out any common factors from the numerator and the denominator of the resulting fraction. Assuming that \(x \neq 2, -2,-3\) (since these make the denominator zero)

Key Concepts

Factoring Quadratic Expressions

When approaching a problem that requires factoring quadratic expressions, our goal is to rewrite the quadratic in a product of binomials. A quadratic expression is something that looks like ax^2 + bx + c. Factoring simplifies complex expressions and can make other algebraic operations, like adding algebraic fractions, much easier.

For instance, when you have a quadratic expression such as x^2 - 4, it's crucial to recognize it as a difference of squares. This specific type can be factored into (x - 2)(x + 2). Another example given in the exercise, x^2 + 5x + 6, factors into (x + 2)(x + 3) since 2 and 3 are numbers that add up to 5 (the middle coefficient) and multiply to 6 (the constant term).

It's essential to be comfortable with factoring as it is a foundational skill in algebra that allows for simplification and solution of equations.

Common Denominator

The concept of a common denominator is used to combine fractions. The common denominator represents a shared multiple of the original denominators. To add or subtract fractions with different denominators, you must first transform them into equivalent fractions with the same denominator.

In our exercise, we see that the denominators are factored into (x-2)(x+2) and (x+2)(x+3). To combine these fractions, we need a common denominator that has all factors from each denominator, which is (x-2)(x+2)(x+3). This may require multiplying the numerators and denominators of the original fractions by any missing factors to ensure they share the common denominator, setting the stage for addition.

Simplifying Algebraic Fractions

Simplifying algebraic fractions, which is reducing them to their simplest form, ensures that the expression is more comprehensible and easier to work with. Once the algebraic fractions share a common denominator and their numerators have been combined, the resulting fraction can often be simplified.

Simplification involves dividing the numerator and denominator by their greatest common factor (GCF). In the exercise, once we add the fractions and combine the numerators, there might be an opportunity to simplify further. However, it is important to note that cancellation can only occur if the factors are present in both the numerator and the denominator.

Equivalent Fractions

Working with equivalent fractions is another important skill in algebra. These are different fractions that represent the same value. To create them, you can multiply or divide the numerator and denominator of a fraction by the same non-zero number.

When you have a fraction, such as x/x^2+5x+6, and you need it to have a different denominator to add it to another fraction, you must find an equivalent fraction with the desired denominator. In the exercise, we manipulate the original fractions so that they have the common denominator, ensuring that they are equivalent in value to the original fractions, just with a new form that allows for addition.