Solution: To find the slant asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $, we perform polynomial long division of $ t^2 + 4 $ by $ t + 2 $:
["Understanding Slant Asymptotes: Finding the Asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $ Using Polynomial Long Division", "When analyzing rational functions like $ I(t) = \frac{t^2 + 4}{t + 2} $, identifying asymptotes helps understand the function’s long-term behavior. One common type is the slant (or oblique) asymptote, which occurs when the degree of the numerator exceeds the degree of the denominator by exactly one. In this case, both the numerator and denominator are polynomials of degree 2 and 1 respectively — the numerator has degree one higher, making a slant asymptote possible.", "### What Is a Slant Asymptote?", "A slant asymptote is a linear function that the original rational function approaches as $ t \ o \infty $ or $ t \ o -\infty $. It is found by performing polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.", "---", "### Step-by-Step Solution: Finding the Slant Asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $", "We divide $ t^2 + 4 $ by $ t + 2 $ using polynomial long division.", "#### Step 1: Set up the division", "$$\n\frac{t^2 + 0t + 4}{t + 2}\n$$", "We divide $ t^2 $ by $ t $ to get the first term of the quotient:\n$ t^2 \div t = t $", "#### Step 2: Multiply and subtract", "Multiply $ t $ by $ t + 2 $:\n$ t(t + 2) = t^2 + 2t $", "Subtract from the original numerator:\n$$\n(t^2 + 0t + 4) - (t^2 + 2t) = -2t + 4\n$$", "#### Step 3: Next division step", "Now divide $ -2t $ by $ t $:\n$ -2t \div t = -2 $", "Multiply $ -2 $ by $ t + 2 $:\n$ -2(t + 2) = -2t - 4 $", "Subtract:\n$$\n(-2t + 4) - (-2t - 4) = 8\n$$", "---", "### Interpreting the Result", "The division yields:\n$$\nI(t) = t - 2 + \frac{8}{t + 2}\n$$", "As $ t \ o \infty $ or $ t \ o -\infty $, the remainder term $ \frac{8}{t + 2} \ o 0 $. Therefore, the function $ I(t) $ approaches the line:", "$$\ny = t - 2\n$$", "---", "### Final Answer", "The slant asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $ is\n$$\n\boxed{y = t - 2}\n$$", "---", "### Why This Matters", "Recognizing the slant asymptote allows us to predict the behavior of $ I(t) $ for large values of $ t $, simplifying graphing and modeling in fields like economics, physics, and engineering. Combined with vertical asymptotes (found where the denominator is zero), polynomial long division provides a complete picture of rational function behavior.", "Keywords: slant asymptote, polynomial division, rational functions, $ \frac{t^2 + 4}{t + 2} $, oblique asymptote, mathematical asymptotes, long division explanation."]
