Problem 44 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x^{2}-4}{x-2}+\frac{x+2}{4 x-8}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(((5/2)x + 5) / (2x - 4)\)

Step 1: Factorize and Simplify

The first step is to factorize the first expression in the numerator of the first fraction. \(x^{2}-4\) can be rewritten as \((x-2)(x+2)\). We also observe that the second fraction \((x+2)/(4x-8)\) can be simplified as well. The expression \(4x-8\) can be factored out as \(4(x-2)\). So the initial problem becomes \((x-2)(x+2) / (x-2) + (x+2) / 4(x-2)\).

Step 2: Cancel Common Factors

The next step is to cancel out common factors that appear in both the numerators and the denominators of the fractions. Here, we can cancel out \(x-2\) from the first fraction, which leaves us with \(x+2\). For the second fraction there are no common factors that can be cancelled out. So the expression now simplifies to \(x+2 + (x+2) / 4(x-2)\).

Step 3: Combine Like Terms

Now combine like terms in order to simplify the algebraic expression. The expression now becomes \((4x+8 + x + 2) / 4(x-2)\).

Step 4: Further Simplify

In the numerator we have \(5x+10\). Dividing each term in the numerator and denominator by 2 for simplification gives \((5/2)x + 5 / 2(x - 2)\), or \(((5/2)x + 5) / (2x - 4)\)

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down complex polynomial expressions into simpler products of its factors. This simplification is essential when dealing with algebraic fractions since it allows us to cancel out common terms.
To factor a polynomial, look for patterns or use methods such as:

• Factoring out the greatest common factor (GCF): This involves dividing each term by the largest factor common to all terms.
• Factoring by grouping: Useful for polynomials with four or more terms. Group the terms to reveal patterns, then factor each group.
• Special factorizations: Recognize patterns like the difference of squares, perfect square trinomials, or sum/difference of cubes. For instance, the difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). In our exercise, \(x^2 - 4\) becomes \((x - 2)(x + 2)\).

These techniques convert complicated expressions into products of simpler terms, easing the way to further simplify and solve algebraic problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with and understand. This often includes factoring, canceling, and expanding expressions by applying algebraic rules to make the math less cumbersome.
In the context of algebraic fractions, simplifying expressions frequently involves canceling out terms that appear in both the numerator and the denominator. Here’s how you can simplify:

• Cancel common factors: After factoring, look for terms that appear both in the numerator and the denominator. Cancel these terms to simplify the fraction. For example, in the first expression of the exercise, after factoring, \(x - 2\) cancels out, simplifying the fraction dramatically.
• Reduce fractions to lowest terms: Similar to numeric fractions, algebraic fractions should be simplified to make expressions easier to handle.

It's important to manipulate expressions until they are in the simplest form to make further operations clearer and more straightforward.

Combining Like Terms

Combining like terms is an essential step when simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. This simplifies the expression and makes it more concise.
Here's how you can combine like terms:

• Identify like terms: Terms are alike if they have the same variable parts. For example, \(x\) and \(3x\) are like terms, while \(x\) and \(x^2\) are not.
• Add or subtract coefficients: Once like terms are identified, add or subtract their coefficients. In the exercise, \(4x + 8\) and \(x + 2\) become \(5x + 10\) when combined.

This process is crucial before applying further algebraic operations, as it simplifies the expressions to their most straightforward form, facilitating easier computation.