Question

Find the sum or difference. \(\frac{3 x^2}{x-8}+\frac{6 x}{x-8}\)

Short answer

The sum of the given fractions is \( \frac{3x^2+6x}{x-8}\).

Step-by-step solution

Identify common denominators

The fractions \(\frac{3 x^2}{x-8}\) and \(\frac{6 x}{x-8}\) both have the same denominator, \(x-8\). So, they can be added together.

Combine the fractions

To add or subtract fractions with the same denominator, combine the numerators over the common denominator. So, sum up \(3x^2\) and \(6x\) and place over \(x-8\) to form \( \frac{3x^2+6x}{x-8}\).

Simplify the fraction as much as possible

In this case, the numerator can't be simplified further. So, the sum of the given fractions is \( \frac{3x^2+6x}{x-8}\).

Key concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like regular numerical fractions, we can perform arithmetic operations such as addition, subtraction, multiplication, and division on them. However, it's crucial to remember that the denominator in a rational expression cannot equal zero as this would make the expression undefined. In our exercise, the rational expressions are \( \frac{3x^2}{x-8} \) and \( \frac{6x}{x-8} \), both of which are defined as long as \( x eq 8 \). This means we should avoid any value of \( x \) that makes the denominator zero. Before performing operations on rational expressions, it's a good idea to factor any possible polynomials in the numerators and denominators to simplify the process later. This helps in easily identifying common factors or making simplifications, if necessary.

Common Denominators

One of the most important steps in handling rational expressions, especially during addition or subtraction, is finding a common denominator. Fractions can only be added or subtracted directly when they share the same denominator. This is crucial because rational expressions must be over an identical baseline to maintain equivalency when combining. In our original exercise, the rational expressions \( \frac{3x^2}{x-8} \) and \( \frac{6x}{x-8} \) already have a common denominator of \( x-8 \). This greatly simplifies the process, as we do not need to perform any additional steps to equate the denominators. If the denominators were different, we would need to find the least common multiple (LCM) of the denominators to create a common basis for combining the fractions.

Polynomial Addition

Adding polynomials involves combining like terms, which are terms having the same variable raised to the same power. When dealing with the addition of rational expressions, this process takes place in the numerator once a common denominator is established. In our exercise situation, the numerator consists of the polynomials \(3x^2\) and \(6x\). Due to the common denominator \( x-8 \), we combine the numerators by simply adding \(3x^2 + 6x\). Though these terms are unlike, meaning they cannot be combined to simplify further, they are united under a single fraction ensuring the expression \( \frac{3x^2+6x}{x-8} \) holds the result of our addition. Polynomial addition is about aligning terms correctly and keeping the denominator in rational expressions constant, which maintains the value and clarity of the final result.