Problem 65 - Algebra 2

Question

Add or subtract. Simplify where possible. $$ \frac{4}{x^{2}-36}+\frac{x}{x-6} $$

Step-by-Step Solution

Answer

\(\frac{x^2 +6x+4}{(x+6)(x-6)}\)

Step 1: Identify the Common Denominator

The denominators from both fractions are \(x^{2}-36\) and \(x-6\). Notice that \(x^{2}-36\) is a difference of squares and can be factored to \((x+6)(x-6)\). Thus, \((x+6)(x-6)\) can be the common denominator.

Step 2: Rewrite the Fractions

Now rewrite each fraction with the common denominator. The first fraction is already with the correct denominator, so we only need to adjust the second one. Rewrite \(\frac{x}{x-6}\) as \(\frac{x(x+6)}{(x-6)(x+6)}\). When multiplying \(x\) with \((x+6)\), it's like distributing \(x\) to both \(x\) and \(6\), so we obtain \(x^2+6x\). So the new form of the fraction is \(\frac{x^2+6x}{(x-6)(x+6)}\).

Step 3: Combine the Fractions

Since both fractions now have the same denominator, they can be added together. Hence, it becomes: \(\frac{4}{(x+6)(x-6)} + \frac{x^2+6x}{(x-6)(x+6)}\). Combining them will give: \(\frac{x^2 +6x+4}{(x+6)(x-6)}\).

Step 4: Simplifying the Result

The final expression \(\frac{x^2 +6x+4}{(x+6)(x-6)}\) cannot be simplified further as there are no common factors that can be cancelled out in the numerator and the denominator. Thus, the expression is already fully simplified.

Key Concepts

Common Denominator

When working with algebraic fractions, it is important to find a common denominator for the expressions you're adding or subtracting. This helps to ensure that the fractions have the same denominator, allowing for straightforward combination. To find a common denominator, you must look at the denominators of each fraction involved.
For example, in the problem we have:

• The denominators are \(x^2-36\) and \(x-6\).
• Recognizing \(x^2-36\) as a difference of squares, it can be rewritten as \((x+6)(x-6)\).

In this case, the common denominator is \((x+6)(x-6)\) because it encompasses both \(x-6\) and the factors \((x+6)(x-6)\). This common denominator is essential for combining fractions correctly, ensuring that the arithmetic operations can be carried out properly.

Factoring Polynomials

Factoring polynomials helps in simplifying expressions and finding common denominators. The process involves rewriting a polynomial as a product of its factors. This can reveal opportunities to simplify expressions or find common factors.
In our example, \(x^2-36\) is a difference of squares, which is a specific type of polynomial expression. It can be factored using the formula:

• \(a^2 - b^2 = (a+b)(a-b)\)

Applying this to \(x^2-36\), you get:

• \(x^2 - 36 = (x+6)(x-6)\)

Recognizing and factoring polynomials in this way helps us to understand their structure and simplify algebraic operations like addition and subtraction of fractions.

Simplifying Expressions

After rewriting fractions using a common denominator, the next step is combining and simplifying the results. Simplifying expressions involves reducing them to their simplest form by either combining like terms or cancelling out terms.
In our exercise:

• We combined the fractions as: \(\frac{4}{(x+6)(x-6)} + \frac{x^2+6x}{(x-6)(x+6)}\).
• The fractions have the same denominator, so you can combine the numerators: \(x^2+6x+4\).

Finally, you check for any common factors in the numerator and denominator to see if further simplification is possible. Here, no such factors exist, so \(\frac{x^2 +6x+4}{(x+6)(x-6)}\) is the simplest form. Simplification not only makes expressions easier to understand but also resolves the overall task of the exercise efficiently.