Polynomial Function
A function of the form f(x) = a_n x^n + ... + a_1 x + a_0. The degree n determines end behavior and the maximum number of zeros.
Example: f(x) = 2x^3 - 5x^2 + 3x - 1 is degree 3 with at most 3 real zeros and 2 turning points.
Adaptive
Learn Polynomial and Rational Functions
Read the notes, then try the practice. It adapts as you go.When you're ready.
Session Length
~13 min
Adaptive Checks
12 questions
Transfer Probes
7
Lesson Notes
Polynomial and rational functions are central to AP Precalculus. Polynomials are built from powers of x; their degree determines end behavior and the maximum number of zeros and turning points.
Rational functions arise when one polynomial is divided by another, introducing vertical asymptotes, horizontal asymptotes, and holes. Mastering zeros, multiplicity, end behavior, asymptotic analysis, and polynomial division prepares students for calculus concepts like limits and continuity.
You'll be able to:
- Analyze polynomial functions using zeros, multiplicity, and end behavior to sketch graphs
- Identify vertical and horizontal asymptotes and holes in rational functions
- Apply the factor theorem and remainder theorem to factor and evaluate polynomials
- Perform polynomial long division and synthetic division
One step at a time.
Interactive Exploration
Adjust the controls and watch the concepts respond in real time.
Key Concepts
Zeros and the Factor Theorem
If f(c)=0, then (x-c) is a factor. Zeros are where the graph crosses or touches the x-axis.
Example: f(x) = x^2 - 5x + 6 has zeros at x=2 and x=3 because f(x) = (x-2)(x-3).
Multiplicity
How many times a factor (x-c) appears. Odd multiplicity: graph crosses the axis. Even multiplicity: graph bounces off.
Example: f(x) = (x-1)^2(x+2) has a double zero at x=1 (bounce) and a single zero at x=-2 (cross).
End Behavior
How f(x) behaves as x approaches positive or negative infinity. Determined by the leading term a_n x^n.
Example: For f(x) = -3x^4 + ..., both ends point down because degree is even and leading coefficient is negative.
Rational Function
A ratio of two polynomials: f(x) = p(x)/q(x). Domain excludes values where q(x) = 0.
Example: f(x) = (x+1)/(x^2-4) has domain all reals except x = 2 and x = -2.
Vertical Asymptote
A vertical line x = a where the function approaches infinity. Occurs where the denominator is zero but the numerator is not.
Example: f(x) = 1/(x-3) has a vertical asymptote at x = 3.
Horizontal Asymptote
The value y = L that f(x) approaches as x goes to infinity. Determined by comparing degrees of numerator and denominator.
Example: f(x) = (2x+1)/(x-3) has HA y = 2 because degrees are equal and leading coefficients ratio is 2/1.
Holes (Removable Discontinuities)
Points where both numerator and denominator are zero due to a common factor. The factor cancels but the point is undefined.
Example: f(x) = (x^2-1)/(x-1) = x+1 with a hole at x=1 (both top and bottom are zero there).
Explore your way
Choose a different way to engage with this topic — no grading, just richer thinking.
Explore your way — choose one:
Concept Map
See how the key ideas connect. Nodes color in as you practice.
Worked Example
Walk through a solved problem step-by-step. Try predicting each step before revealing it.
Adaptive Practice
This is guided practice, not just a quiz. Hints and pacing adjust in real time.
Small steps add up.
What you get while practicing:
- Math Lens cues for what to look for and what to ignore.
- Progressive hints (direction, rule, then apply).
- Targeted feedback when a common misconception appears.
Ad Placeholder
Teach It Back
The best way to know if you understand something: explain it in your own words.
Keep Practicing
More ways to strengthen what you just learned.