Problem 56 - College Algebra

Question

Exercises \(55-57\) will help you prepare for the material covered in the next section. $$ \text { Add }: \frac{5 x-3}{x^{2}+1}+\frac{2 x}{\left(x^{2}+1\right)^{2}} $$

Step-by-Step Solution

Answer

The solution is \(\frac{5x^{3}-3x^{2}+7x-3}{(x^{2}+1)^{2}}\).

Step 1: Analyze the Denominators

Notice that both denominators are powers of \(x^{2}+1\). This means we must bring both fractions to the same power of \(x^{2}+1\) to add them properly. The common denominator will be \((x^{2}+1)^{2}\)

Step 2: Make Denominators Equal

For the first fraction \(\frac{5x-3}{x^{2}+1}\), multiply the numerator and the denominator by \(x^{2} + 1\) to obtain the common denominator. This gives us: \[\frac{(5x-3)(x^{2}+1)}{(x^{2}+1)(x^{2} + 1)}\] which simplifies to \[\frac{5x^{3}+5x-3x^{2}-3}{(x^{2}+1)^{2}}\] which can further be simplified to \[\frac{5x^{3}-3x^{2}+5x-3}{(x^{2}+1)^{2}}\]

Step 3: Add the Fractions

Now that both fractions have the same denominator, they can be added. Let's add \(\frac{5x^{3}-3x^{2}+5x-3}{(x^{2}+1)^{2}}\) and \(\frac{2x}{(x^{2}+1)^{2}}\). This gives us: \[\frac{5x^{3}-3x^{2}+5x-3+2x}{(x^{2}+1)^{2}}\] Collect like terms to get the final expression: \[\frac{5x^{3}-3x^{2}+7x-3}{(x^{2}+1)^{2}}\]

Key Concepts

Common Denominators in Algebra

When dealing with algebraic fractions, finding common denominators is crucial for performing operations like addition and subtraction. Fractions can only be added or subtracted directly when they share the same denominator. Just as you would find a common denominator when adding simple numerical fractions like \frac{1}{3} and \frac{1}{6}, with algebraic fractions, you must also find a denominator that both terms can share.

In the exercise, the denominators are powers of \(x^2+1\). To combine these fractions, we need to manipulate them so they have the same denominator, in this case, \(x^2 + 1)^2\). This involves multiplying the numerator and the denominator of the first fraction by \(x^2+1\) to match the denominator of the second fraction. By doing this, we ensure a consistent denominator across both terms, allowing us to combine them properly in the subsequent step.

Simplifying Algebraic Fractions

Once we've established a common denominator, the next step is simplifying algebraic fractions, which involves reducing expressions to their simplest form to make operations more straightforward. In our exercise, after achieving the common denominator, the next move is to simplify the new numerators.

When multiplying out the numerators, you perform polynomial multiplication, which, although not as simple as multiplying numbers, follows a similar distributive principle. The exercise requires multiplying \(5x - 3\) by \(x^2 + 1\), distributing each term in the first polynomial across each term in the second. After this, combine like terms in the result to get a simplified numerator. The final step is to combine the numerators of the aligned fractions, ensuring that like terms are collected to achieve the most simplified form of the algebraic expression.

Polynomial Arithmetic in Algebra

Arithmetic with polynomials includes operations such as addition, subtraction, multiplication, and division. It is a foundation for working with algebraic fractions. The key to working with polynomials is recognizing that they are sums of terms with variables raised to various powers and coefficients.

In the provided exercise, after simplifying the individual algebraic fractions, you undertake polynomial addition. To add polynomials, one combines like terms — terms that have the same variable raised to the same power. For example, in the final step of our solution, we added \(5x^{3} - 3x^{2} + 5x - 3\) and \(2x\), combining the like terms \(5x\) and \(2x\) to end up with \(7x\), thus giving us \(5x^{3} - 3x^{2} + 7x - 3\), all over the common denominator \(x^{2}+1)^2\). Through these steps of polynomial arithmetic, the initial complex expression is reduced to a more manageable form.