Polynomial Functions
Unlock the secrets of higher-degree equations. Master the Remainder Theorem and graph behaviors to solve complex SAT problems in seconds.
The Remainder Theorem
The SAT often asks for the remainder when a polynomial \(p(x)\) is divided by a linear factor \((x - c)\). You don't need long division! Just plug in the value:
Fast Hack
If the question says "\(p(x)\) is divided by \((x + 2)\)", calculate \(p(-2)\).
Factor Connection
If \(p(c) = 0\), then \((x - c)\) is a factor of the polynomial.
Graph Characteristics
Recognizing polynomial graphs based on their degree and leading coefficient is a high-value SAT skill.
Zeros and Multiplicity
- • Standard Root: Graph crosses the x-axis.
- • Double Root (Squared): Graph "bounces" off the x-axis.
- • Triple Root (Cubed): Graph "flattens" as it crosses.
Turning Points
A polynomial of degree \(n\) can have at most \(n - 1\) turning points (peaks and valleys).
⚠️ The Long Division Trap
Don't waste time on full polynomial long division unless the answer choices are in quotient form. For "finding the remainder" or "checking factors," the **Remainder Theorem** is always 10x faster.