AP Precalculus: Polynomial Functions Formulas
1. Roots / Zeros of Factored Polynomials
If \( f(x) = a(x-r_1)(x-r_2)\cdots(x-r_n) \), then roots/zeros: \( x = r_1, r_2, \ldots, r_n \)
To find roots, set each factor to zero: \( x-r_k=0 \rightarrow x=r_k \)
2. Write Polynomial from Roots
Given roots \( r_1, ..., r_n \),
General form: \( f(x) = a(x - r_1)(x - r_2)\ldots(x - r_n) \)
3. Rational Root Theorem
Possible rational roots: \( \pm \frac{\text{factors of constant}}{\text{factors of leading coefficient}} \)For \( a_nx^n + \ldots + a_0 \): try \( \pm \) all fractions \( \frac{p}{q} \) where \( p \mid a_0 \), \( q \mid a_n \)
4. Complex Conjugate Theorem
If \( a+bi \) is a root and all coefficients are real, \( a-bi \) is also a root.5. Conjugate Root Theorem (Irrational)
If \( a+\sqrt{b} \) is a root and coefficients are rational, then \( a-\sqrt{b} \) is also a root.6. Descartes' Rule of Signs
- Number of positive real roots = number of sign changes in \( f(x) \) or less by a multiple of 2
- Number of negative real roots = number of sign changes in \( f(-x) \) or less by a multiple of 2
7. Fundamental Theorem of Algebra
Every n-th degree polynomial (with complex coefficients) has exactly n complex roots (counting multiplicities).8. End Behavior (Leading Term Test)
As \( x \to \infty \):
if degree even, both ends match leading coefficient; if degree odd, ends are opposite.
e.g. For \( f(x) = ax^n \): if \( a > 0 \), ends go up for even; if \( a < 0 \), ends go down for even, or left up/right down for odd.
9. Matching Polynomials and Graphs Using Zeros
- Zero/root = x-intercept
- Multiplicity: even = touch; odd = cross at x-axis
- Ghost factors (no root): disappear
10. Domain & Range
- Domain: all real numbers (\( (-\infty, \infty) \))
- Range: depends on degree and leading coefficient
11. Even and Odd Functions
- Even: \( f(-x) = f(x) \) (symmetric about y-axis)
- Odd: \( f(-x) = -f(x) \) (symmetric about origin)
- Polynomials can be even, odd, or neither