Summary Basic Notes on Polynomials for Precalculus

1. Formal Definition & Constraints
A polynomial function of degree n is defined as:

P(x)=an​xn+an−1​xn−1+⋯+a1​x+a0​

●​ The Rules: n must be a non-negative integer (no x1/2 or
x−2). The coefficients a must be real numbers.
●​ Leading Coefficient (an​): The number attached to the
highest power. It dictates the "stretch" and the right-hand
end behavior.
●​ Degree (n): The highest exponent. It tells you the
maximum number of x-intercepts (n) and the maximum
number of turning points (n−1).

2. The Leading Coefficient Test (End Behavior)
As x→∞ or x→−∞, the polynomial mimics its leading term
(an​xn).

Degree Leading Left End Right End
(n) Coeff (an​) (x→−∞) (x→∞)
Even Positive (+) ∞ (Rises) ∞ (Rises)
Even Negative (-) −∞ (Falls) −∞ (Falls)
Odd Positive (+) −∞ (Falls) ∞ (Rises)
Odd Negative (-) ∞ (Rises) −∞ (Falls)

Pro-Tip: If the degree is Even, the ends go in the
Same direction (both up or both down). If the degree is
Odd, the ends go in Opposite directions.

, 1. The Equivalent Statements
If c is a real number, the following are functionally identical:

1.​x=c is a zero (or root) of P(x).
2.​P(c)=0.
3.​(x−c) is a linear factor of P(x).
4.​(c,0) is an x-intercept of the graph.

2. Multiplicity and Graphing "Shapes"
If a factor appears multiple times, (x−c)k, the exponent k is the
multiplicity:

●​ Multiplicity 1: The graph passes straight through the
x-axis (linear look).
●​ Even Multiplicity (2, 4, ...): The graph touches the axis
and bounces back (parabolic look).
●​ Odd Multiplicity > 1 (3, 5, ...): The graph flattens as it
passes through (cubic look).

3. Intermediate Value Theorem (IVT)
If P(x) is a polynomial and P(a) and P(b) have opposite signs
(one is positive, one is negative), then there is at least one
value c between a and b such that P(c)=0.

●​ Use Case: Use this to prove a root exists in a specific
interval (e.g., between x=1 and x=2).