Question

Express as a single fraction $$ s+2+\frac{s+3}{(s+1)(s+2)} $$

Short answer

Question: Express the given expression as a single fraction: $$s+2+\frac{s+3}{(s+1)(s+2)}$$ Answer: $$\frac{s^3 + 5s^2 + 7s + 7}{(s+1)(s+2)}$$

Step-by-step solution

Identify the common denominator

We have three terms in the expression: $$s$$, $$2$$ and $$\frac{s+3}{(s+1)(s+2)}$$. The denominators of $$s$$ and $$2$$ are $$1$$. So, the common denominator for all terms will be the lowest common multiple (LCM) of their denominators. In this case, the LCM is $$(s+1)(s+2)$$.

Convert terms to have the common denominator

Now, we will convert each term to have the common denominator $$(s+1)(s+2)$$. To convert $$s$$: $$ s \times \frac{(s+1)(s+2)}{(s+1)(s+2)} = \frac{s(s+1)(s+2)}{(s+1)(s+2)} $$ To convert $$2$$: $$ 2 \times \frac{(s+1)(s+2)}{(s+1)(s+2)} = \frac{2(s+1)(s+2)}{(s+1)(s+2)} $$ No need to convert the third term since it already has the common denominator.

Combine the terms with the common denominator

Now that all terms have the common denominator, we will combine them into a single fraction: $$\frac{s(s+1)(s+2)}{(s+1)(s+2)} + \frac{2(s+1)(s+2)}{(s+1)(s+2)} + \frac{s+3}{(s+1)(s+2)} $$

Simplify the fraction

Add the numerators together: $$\frac{s(s+1)(s+2) + 2(s+1)(s+2) + s+3}{(s+1)(s+2)} $$ Now, expand and simplify the numerator: $$\frac{s^3 + 3s^2 + 2s + 2s^2 + 4s + 4 + s + 3}{(s+1)(s+2)} $$ Combine like terms: $$\frac{s^3 + 5s^2 + 7s + 7}{(s+1)(s+2)} $$

Final Answer

So the given expression can be expressed as a single fraction: $$ s+2+\frac{s+3}{(s+1)(s+2)} = \frac{s^3 + 5s^2 + 7s + 7}{(s+1)(s+2)} $$

Key concepts

Fractions

Fractions are a way to express parts of a whole number, using a numerator and a denominator. The numerator is the top part, showing how many parts we have. The denominator is the bottom part, showing how many equal parts the whole is divided into. In algebra, fractions can involve variables, like \( \frac{s+3}{(s+1)(s+2)} \), where both the numerator and the denominator are polynomials. It's important to understand fractions, as they often appear in algebraic expressions. In our exercise, turning each term into a fraction with a common denominator allows us to combine them into a single fraction, following the rules of arithmetic with fractions.
Fractions in math are fundamental, and knowing how to manipulate them is key to solving many algebraic problems, including adding and subtracting terms with different denominators.

Polynomial

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, \( x^3 + 2x^2 + 3x + 4 \) is a polynomial of degree 3. Each part of a polynomial separated by a plus or minus sign is called a term. In the given exercise, expressions like \( s+2 \) and \( s+3 \) come into play.
When dealing with polynomials, we often need to add, subtract, or multiply these expressions. This requires careful attention to like terms, which are terms that have the same variable raised to the same power. Combining like terms simplifies the polynomial, making it easier to work with. For instance, in the simplification step of our problem, combining like terms is crucial to arrive at the simpler form of the polynomial fraction.

Common Denominator

Finding a common denominator is necessary when adding or subtracting fractions. A common denominator is a shared multiple of the denominators of the fractions you are working with. Having a common denominator allows you to combine fractions directly by adding or subtracting their numerators. In our exercise, the common denominator for all terms is \( (s+1)(s+2) \).

• The first step involves determining what the common denominator should be. This is typically done by finding the least common multiple (LCM) of the denominators involved.
• The next step is to convert each term to have this common denominator by multiplying both the numerator and the denominator of the fraction.

Once the fractions share this common multiple, they can be added together easily, as seen in the problem, where each term was rewritten to share the common denominator \( (s+1)(s+2) \).
Understanding how to find and use a common denominator is crucial for combining fractions efficiently.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more concise or easier-to-understand form. For algebraic expressions involving fractions and polynomials, this usually means combining like terms and reducing any complex expressions. In the exercise, simplifying is the final step after combining the fractions with a common denominator.

• The process often starts by expanding any factored expressions in the numerator or denominator.
• The next step is to combine like terms in the expression, which involves adding or subtracting them.
• The goal is to end up with a simpler polynomial or fraction that is easier to interpret or further manipulate.

Simplifying expressions not only makes them neater but also reveals important properties, such as intercepts and asymptotes in a polynomial function. In our example, after combining the numerators, we simplify the expression \( s^3 + 5s^2 + 7s + 7 \), ensuring it presents the clearest solution to the problem.