Chapter 5: Problem 32

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

Short Answer

Expert verified

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: $$ s+2+\frac{s+3}{(s+1)(s+2)} = \frac{s^3 + 5s^2 + 7s + 7}{(s+1)(s+2)} $$

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

Short Answer

Step by step solution

Identify the common denominator

Convert terms to have the common denominator

Combine the terms with the common denominator

Simplify the fraction

Final Answer

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