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

S(t) = rac{t^2 + 5t + 6}{t + 2}

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

Continue Reading

Let $A = (0, 0)$, $C = (3, 4)$.
Midpoint of $AC$:
M_2 = \left( \frac{0 + 3}{2}, \frac{0 + 4}{2} \right) = \left( \frac{3}{2}, 2 \right)

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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: