AP Precalculus -- Polynomial and Rational Functions

Learning objectives

• Find zeros of polynomials by factoring, including grouping and special patterns
• Apply the factor theorem and remainder theorem to verify roots
• Determine how multiplicity affects graph behavior at zeros (cross at odd, bounce at even)
• Use sign charts to determine intervals where a polynomial is positive or negative
• Write a polynomial function given its zeros and their multiplicities

Learning objectives

• Predict end behavior from degree and leading coefficient using limit notation
• Sketch polynomial graphs using zeros, multiplicity, and end behavior together
• Determine the maximum number of turning points from the degree of the polynomial
• Analyze how transformations (shifts, reflections, stretches) affect polynomial graph shape
• Match polynomial equations to their graphs and vice versa

Learning objectives

• Identify vertical asymptotes from denominator zeros that do not cancel
• Determine horizontal and slant asymptotes by comparing degrees of numerator and denominator
• Distinguish between vertical asymptotes and holes (removable discontinuities)
• Graph rational functions using intercepts, asymptotes, and test points
• Determine the domain of a rational function and express it in interval notation

Learning objectives

• Perform polynomial long division and express the result as quotient plus remainder over divisor
• Apply synthetic division for linear divisors and interpret the result
• Use the remainder theorem to evaluate polynomials quickly without full division
• Select appropriate polynomial regression models for data sets using residual analysis
• Explain when and why a higher-degree polynomial model may overfit a dataset