Polynomial Functions in a Nutshell
Definition: A polynomial of degree n is a function (𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 . The
numbers 𝑎𝑗 are called coefficients, the term with the highest power 𝑎𝑛 is called the leading coefficient,
and the number 𝑎0 is the constant coefficient.
Zeros and Extrema: A polynomial of degree n can have


at most n zeros
at most n-1 max or mins
Intermediate Value Theorem: If p(x) is a polynomial and a and b are two real numbers such that p(a)
and p(b) have different signs. Then there is a number c between a and b such that p(c) = 0
Remainder Theorem: If you divide a polynomial of degree n (via long division) by (x – c) the remainder is
p(c).
Factor Theorem: If a polynomial p(c) = 0, then p is divisible by (x – c)
Rational Zeros Theorem: If p(x) is a polynomial with rational coefficients, and if it has any rational zero
p/q then p divides the constant coefficient of p(x) and q divides the leading coefficient.
Descartes Rule of Signs: If p(x) is a polynomial such that:


if p(x) has n sign changes, then p has n or n-2 or n-4 etc. positive zeros
p(-x) has m sign changes, then p has n or n-2 or n-4 etc. negative zeros
Rational Functions in a Nutshell
𝑝(𝑥)
Definition: A rational function is the quotient of two polynomials: 𝑟(𝑥) = 𝑞(𝑥)
Zeros: 𝑟(𝑥) =
𝑝(𝑥)
𝑞(𝑥)
= 0 if and only if the top polynomial p(x) = 0
Domain: Set bottom polynomial equal to zero and solve. Domain is all real numbers except for q(x) = 0
Asymptotes: the distinguishing feature of rational function. There are two kinds:
Vertical: Solve q(x) = 0 (i.e. each number not in the domain). Check if as x gets close to an
asymptote from the left and the right, the rational function gets close to ±infinity.
Vertical asymptotes have the form x = c, and most rational functions (but not all) have one or more.
Horizontal: y = c, where c is the limit of r(x) as x goes to ±infinity. Rational functions have at most
one of these and many have none. The limit c depend on the degree of top and bottom polynomial:
y = 0: if degree of the top polynomial is less than the degree of the bottom one
y = c: if degree of top is the same as the degree of the bottom, and the number c is the quotient
of the leading coefficients of p(x) and q(x)
none: if degree of the top is greater than the degree of the bottom
Graph: To graph a rational function, follow these steps:
1. Solve q(x) = 0, i.e. find all vertical asymptotes.
For each number you find, check the sign of
the function when you are to the left or right
of your number. Draw the vertical lines, if any,
and indicate on which side the graph will go to
infinity.
2. Find the horizontal asymptote by finding the
limit as x goes to infinity. If there is a
horizontal asymptote, draw it into your
coordinate system.
3. Mark all x-intercepts by setting the top q(x) = 0
4. Find the y-intercept p(0)
5. Try to fit the parts of the graph between
asymptotes to the above info