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:

📰 wolverine crossing 📰 the met apartments 📰 apartments dallas tx 📰 Center Grove Softball 6354867 📰 Video Scheduler Sounds Like A Disaster Internal Error Is Sabotaging Your Workflow 4951486 📰 Unlock The Secret To Maximum 2024 Hsa Savings Do Not Miss This Limit 6330175 📰 Master Primavera P6 Like A Champion Top Rated Course Worth Your Time And Money 1043354 📰 This Kiss In The Kiss Cartoon Changed Everything Forever 673997 📰 How To Make A Copper Golem In Minecraft 9499356 📰 A Cylindrical Tank With A Radius Of 3 Meters And A Height Of 5 Meters Is Filled With Water Calculate The Volume Of The Water In Cubic Meters 6562243 📰 The Impossible Return Of Kuro One Piece Viewers Are Obsessed Shocked 7297837 📰 5 Watch Live Channel 7 News Buffalo Captures National Attention Tonight 1772382 📰 Centennial Airport 8381634 📰 Unlock The Booklet Aesthetic Secrets That Will Trans Correct Your Style Forever 1509331 📰 Verizon Iphone Sos Mode Today 5180654 📰 Robert Taylor American Actor 3627899 📰 Wells Fargo Coconut Creek Fl 4511500 📰 Trump Son Sees Father At Thanksgivingunlikely Moment Devastates Social Media 3093110

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: