Problem 60 - Algebra and Trigonometry

Question

Add: \(\frac{5 x-3}{x^{2}+1}+\frac{2 x}{\left(x^{2}+1\right)^{2}}\)

Step-by-Step Solution

Answer

The sum of the fractions is \( \frac{2x^{3}+7x-3}{(x^{2}+1)^{2}} \)

Step 1: Analyse the fractions

Start by looking at the two fractions. They both have \(x^{2}+1\) in their denominator, but the second fraction's denominator is squared, while the first fraction's denominator isn't.

Step 2: Multiply second fraction to match first fraction's denominator

Multiply the numerator and the denominator of the second fraction by \(x^{2}+1\). So, \(\frac{2x}{(x^{2}+1)^{2}}\) becomes \(\frac{2x(x^{2}+1)}{(x^{2}+1)^{2}}\), and finally \(\frac{2x^{3}+2x}{(x^{2}+1)^{2}}\).

Step 3: Add the fractions

Now that the denominators are the same, the fractions can be added together. \(\frac{5x-3}{x^{2}+1} + \frac{2x^{3}+2x}{(x^{2}+1)^{2}} = \frac{2x^{3}+2x+5x-3}{(x^{2}+1)^{2}}\).

Step 4: Rearrange the terms

Rearrange the terms in the numerator to give: \(\frac{2x^{3}+7x-3}{(x^{2}+1)^{2}}\)

Key Concepts

Common Denominator

When adding rational expressions, one crucial step is to find a common denominator. A common denominator is the same value in the denominator for all the expressions in a problem.
To successfully add fractions, they should share this common denominator.
In our exercise, we have two rational expressions with different denominators:

• The first expression, \( \frac{5x-3}{x^2+1} \), has the denominator \( x^2+1 \).
• The second expression, \( \frac{2x}{(x^2+1)^2} \), features the denominator \( (x^2+1)^2 \).

To align these, we need to make the denominators identical. In this problem, multiplying the first denominator and its corresponding numerator by \( x^2+1 \) accomplishes that. Once the denominators match, you can easily add the expressions together.

Polynomial Addition

Polynomial addition involves summing the coefficients of like terms in a numerator once the denominators are the same.
In our exercise, after adjusting the rational expressions to have a common denominator, the next step is combining the numerators.Here's how you proceed:

• The adjusted polynomial expression from the second fraction becomes \( 2x^3 + 2x \).
• Add this to the first polynomial expression, \( 5x - 3 \).

The resulting expression in the numerator is obtained by combing like terms: \( 2x^3 + 2x + 5x - 3 \). It then simplifies to \( 2x^3 + 7x - 3 \).
The key to polynomial addition is combining terms with the same degree. This helps in maintaining clear and organized expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials.
Working with rational expressions requires a clear understanding of both polynomials and fractions because they share characteristics of each.
These expressions can involve complex operations, such as addition, subtraction, multiplication, or division, similar to how you handle numerical fractions. In this case, the rational expressions added are:

• \( \frac{5x-3}{x^2+1} \)
• \( \frac{2x}{(x^2+1)^2} \)

Handle these expressions using the same rules applicable to numeric fractions, including finding a common denominator, simplifying, and combining like terms.

Fraction Simplification

Fraction simplification involves making an expression as simple as possible. This can be achieved by factoring the numerator and the denominator and then reducing common factors if they exist.
In the provided exercise, once you've aligned the denominators and added the numerators, consider simplification as the final step: For the resulting expression \( \frac{2x^3 + 7x - 3}{(x^2+1)^2} \), look to see if the numerator or denominator can be simplified further:

• Check for any common factors that are in both the numerator and denominator.
• Factor where possible to reduce the expression to its simplest form.

This process helps in creating a more manageable expression that is easier to interpret and use, especially for further mathematical computations.