Lesson 4.3

The Leading Term Test

In the long run, only one term matters. How to predict the future of any polynomial.

Introduction

In the long run, only one term matters. How to predict the future of any polynomial.

Past Knowledge

In Lesson 4.2, we saw that odd powers go "Down-Up" or "Up-Down", while even powers go "Up-Up" or "Down-Down". This lesson proves that complex polynomials behave exactly like their simpler power function parents when is large.

Today's Goal

The Leading Term Test allows us to ignore everything except the term with the highest exponent. behaves just like in the long run.

Future Success

Finding limits at infinity () is literally applying the Leading Term Test. This is the foundation of asymptotic analysis.

Key Concepts

The 4 Possible End Behaviors

Even Degree (+)
↑ ... ↑
Ends both go UP
Like
Even Degree (-)
↓ ... ↓
Ends both go DOWN
Like
Odd Degree (+)
↓ ... ↑
Down Left, Up Right
Like
Odd Degree (-)
↑ ... ↓
Up Left, Down Right
Like

Worked Examples

Level: Basic

Example 1: Identifying the Leader

Determine end behavior for .

Step 1: Find Leading Term
It's .
Step 2: Analyze Degree and Sign
  • Degree 4 is EVEN.
  • Coefficient 3 is POSITIVE.
Answer
Up on both sides ( as ).
Level: Intermediate

Example 2: Hidden Leading Term

Describe end behavior of .

Don't let the order fool you! The leading term is .
It is ODD and NEGATIVE.
Start HIGH (left), End LOW (right).
Level: Advanced

Example 3: Summing Degrees

Find end behavior for .

Do NOT expand the whole thing. Just multiply the leading terms of each factor.
Term 1: -2x
Term 2: (x)^2 → x^2
Term 3: (x) → x
Total:
Analysis
is Even and Negative.
The graph points DOWN on both sides.

Common Pitfalls

  • Confusing Coefficient Sign with Exponent Sign:

    is not a polynomial (it's ).
    is a polynomial.
    Only look at the sign of the number in front (coefficient), not the power!

  • Adding Degrees in Non-Factored Form:

    In , the degree is 5.
    In , the degree is 8.
    Know when to pick the winner vs. when to combine them.

Real-Life Applications

Computer Science: Algorithm Complexity

Big O Notation ( vs ) describes how code performs as data sets grow huge.

If an algorithm takes milliseconds, we ignore the 1000 and the 50n. Why? Because when n is a billion, the term is the only one that matters. This is the Leading Term Test in action!

Practice Quiz

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