Question

Express as a single fraction $$ \frac{a_{1}}{s+p_{1}}+\frac{a_{2}}{s+p_{2}} $$

Short answer

Question: Express the sum of the given fractions as a single fraction: $$\frac{a_{1}}{s+p_{1}} + \frac{a_{2}}{s+p_{2}}$$ Answer: The sum of the given fractions expressed as a single fraction is: $$\frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})}$$

Step-by-step solution

Identify the denominators

Identify the denominators of the given fractions. In this case, they are \((s + p_1)\) and \((s + p_2)\).

Find a common denominator

To find a common denominator, we can multiply the two denominators together, as this will always be a multiple of both denominators. So, in this case, the common denominator will be \((s+p_1)(s+p_2)\).

Convert fractions to their equivalents

Now that we have a common denominator, we can convert the fractions into their equivalents with the same denominator. We do this by multiplying the numerator and denominator of each fraction by the necessary factor: $$ \frac{a_{1}}{s+p_{1}}\times\frac{s+p_{2}}{s+p_{2}} = \frac{a_{1}(s+p_{2})}{(s+p_{1})(s+p_{2})} \\ \frac{a_{2}}{s+p_{2}}\times\frac{s+p_{1}}{s+p_{1}} = \frac{a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$

Add the equivalent fractions

Now that the fractions have the same denominator, we can add them together by adding their numerators: $$ \frac{a_{1}(s+p_{2})}{(s+p_{1})(s+p_{2})} + \frac{a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} = \frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$

Final answer

The sum of the given fractions expressed as a single fraction is: $$ \frac{a_{1}(s+p_{2})+a_{2}(s+p_{1})}{(s+p_{1})(s+p_{2})} $$

Key concepts

Fractions

Fractions are an essential concept in algebra and mathematics in general. They represent parts of a whole. In other words, a fraction is a way to divide any quantity into equal-sized pieces. In algebra, you often see fractions with variables in the numerator or denominator, which prepare you for more complex equations, like rational expressions. Fractions consist of two main parts:

• A numerator (the top number), which indicates how many parts are considered.
• A denominator (the bottom number), which shows the number of equal parts the whole is divided into.

Remember, fractions that share the same denominator can be easily added or subtracted by handling their numerators. Acting on fractions requires understanding how to manipulate these parts correctly.

Common Denominator

A common denominator is crucial when you need to add or subtract fractions. It is essentially a common multiple of the denominators involved in your fractions. Having a shared denominator allows for straightforward addition or subtraction.To find a common denominator for \(rac{a_{1}}{s+p_{1}}\) and \(rac{a_{2}}{s+p_{2}}\), one can multiply the two denominator expressions: \((s+p_{1})(s+p_{2})\). This multiplication ensures each individual fraction aligns under a single denominator, setting the stage for their numerators' combination. It's important to note that while multiplying denominators gives a valid common denominator, sometimes you can find simpler ones if you're dealing with numbers instead of algebraic expressions. Nevertheless, multiplying existing denominators is a foolproof method to proceed with algebraic fractions.

Rational Expressions

Rational expressions resemble fractions but involve polynomials both in the numerator and the denominator. These are central in algebra and require mastering similar skills used for simple numerical fractions. When handling rational expressions:

• Focus on finding equivalent expressions that allow for easy operations like addition or subtraction.
• Simplification often involves factoring polynomials to reduce the complexity of the expression.

To compose a single rational expression from multiple terms like in our exercise:

• First, find a common denominator.
• Then adjust each fraction's numerator so that it’s equivalent based on this shared denominator.

Understanding rational expressions lays a strong foundation for delving into calculus topics and rational function analysis.

Simplifying Fractions

Simplifying fractions or rational expressions is the process of reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common factor, or in the case of polynomials, factoring out common terms. Why simplify?

• It makes further calculations more straightforward.
• It presents the expression in its easiest-to-understand format.

In the given exercise, \(\frac{a_{1} (s+p_{2}) + a_{2} (s+p_{1})}{(s + p_{1})(s + p_{2})}\) represents a fully integrated, single fraction of the original terms. While more common in numerical fractions, algebraic simplification isn’t always possible unless common factors exist. Thus, recognizing when fractions are as simple as possible is key to ensuring equations and expressions are correctly handled in algebra.