AP Calculus BC

Polynomial and Rational Functions

Polynomials in factored form

A polynomial written as a product of factors, like $f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}\cdots$, shows its zeros directly: $f(r_i) = 0$ for each factor.

• The degree is the sum of the exponents.
• The leading coefficient is $a$ times the leading coefficient of each factor.
• End behavior depends on the degree's parity and the leading coefficient's sign: an even degree sends both ends the same direction, an odd degree sends them opposite directions, and a positive leading coefficient rises to the right.
• At a zero $r_i$, the graph crosses the x-axis if $m_i$ is odd, and bounces off it (touches and turns back) if $m_i$ is even.

Rational functions

A rational function that has the same factor in its numerator and denominator has a hole there instead of a zero or asymptote, once that factor is cancelled. After cancelling:

• A remaining denominator factor equal to 0 gives a vertical asymptote.
• A remaining numerator factor equal to 0 gives a zero.
• If the numerator and denominator have the same degree, the horizontal asymptote is the ratio of their leading coefficients.

Finding zeros from a known root

If $x = r$ is a known zero of a cubic $j(x)$, the factor theorem says $x - r$ divides $j(x)$ evenly, with no remainder, leaving a quadratic factor. That quadratic's two zeros are either both real, or a complex conjugate pair, depending on its discriminant.