AP Precalculus December '25 Exam (TERMS TO KNOW)

The Remainder Theorem

If a polynomial f(x) is divided by x - k, then the remainder is

r = f(k)

The Factor Theorem

A polynomial f(x) has a factor (x - k) if and only if f(k) = 0

What information does the remainder r provide?

1. The remainder r gives the value of f at x = k. That is, r = f(k)

2. If r = 0, then (x - k) is a factor of f(x)

3. If r = 0, then (k, 0) is an x-intercept of the graph of f

The Rational Zero Test

If the polynomial f(x) = a_nxⁿ+a_n-1x^n-1+ . . . + a₂x² + a₁x + a₀ has integer coefficients, then every rational zero of f has the form

Rational zero = p/q (no common factors other than 1, p is factor of constant term a₀, and q is factor of leading coefficient a_n

Decartes’s Rule of Signs

Let f(x) = a_nxⁿ + a_n-1x^n-1 + . . . + a₂x² + a₁x + a₀ be a polynomial with real coefficients and a₀ ≠ 0.

1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer

2. The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer
   

MISSING TERMS (coefficient 0) CAN BE IGNORED

imaginary unit i

i = √-1

i² = -1

complex number

a number that combines a real part and an imaginary part

The Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system

Linear Factorization Theorem

If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors

Complex Zeros Occur in _____________

Conjugate Pairs