Simplifying the Rational Expression: S(t) = (t² + 5t + 6)/(t + 2)

In algebra, rational expressions are essential tools for modeling polynomial relationships, and simplifying them can make solving equations and analyzing functions much easier. One such expression is:

S(t) = (t² + 5t + 6)/(t + 2)

Understanding the Context

This article explores how to simplify and analyze this rational function, including steps to factor the numerator, check for domain restrictions, and express S(t) in its simplest form.

Step 1: Factor the Numerator

The numerator is a quadratic expression:
t² + 5t + 6

Image Gallery

$Solved: 2. Determine s'(t) for s(t)=3t^(frac 1)3+2t^2 c. a. s'(t)=4t ...$

Solved Find the derivativeddt6(t2+7t)-2ddt6(t2+7t)-2= | Chegg.com

Key Insights

To factor it, look for two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3.

So,
t² + 5t + 6 = (t + 2)(t + 3)

Now rewrite S(t):
S(t) = [(t + 2)(t + 3)] / (t + 2)

Step 2: Simplify the Expression

🔗 Related Articles You Might Like:

📰 The Ultimate Guide: Where to Find Windows Product Keys Without Paying a Dime! 📰 You Wont Believe Where You Can Play Sprunki—Its SIMPLY Unbelievable! 📰 The #1 Secret Place to Play Sprunki? Its Shockingly Under the Radar! 📰 Untied Airlines 8631835 📰 What Are The Cheapest Cell Phone Plans 8095809 📰 Moto M3X Unleashed The Ultimate Ride Thats Setting Motorsports On Fire 5852919 📰 Best Debit Cards 5747192 📰 Activate Fidelity Debit Card Like A Pro Tips That Data Show Works Every Time 780851 📰 A Ensuring All Exhibits Are Technically Complex To Showcase Expertise 803704 📰 Eggman Game 2053823 📰 Townhomes For Rent St Petersburg Fl 4624468 📰 Air Talk Wireless The Secret To Crystal Clear Wireless Conversations 2068291 📰 Upside Down Pineapple Means More Than You Thinkwhat It Reveals Is Shocking 9855148 📰 Arcadia University 7437358 📰 What Exactly Is A Short Message The Shocking Truth Revealed 7530960 📰 No Nonsense Neutering 786058 📰 How Long Does It Take Personal Checks To Clear 8024595 📰 Youtube Tv News 1220154

Final Thoughts

Since (t + 2) appears in both the numerator and the denominator, as long as t ≠ -2, we can cancel this common factor:

S(t) = t + 3, for t ≠ -2

This simplification is valid because division by zero is undefined. So, t = -2 is excluded from the domain.

Understanding the Domain

From the original function, the denominator t + 2 is zero when t = -2. Thus, the domain of S(t) is:
All real numbers except t = -2
Or in interval notation:
(-∞, -2) ∪ (-2, ∞)

Graphical and Analytical Insight

The original rational function S(t) is equivalent to the linear function y = t + 3, with a hole at t = -2 caused by the removable discontinuity. There are no vertical asymptotes because the factor cancels entirely.

This simplification helps in understanding behavior such as: