ƒ(x) = **a**_{n}x^{n} + a_{n-1}x^{n-1} +... + a_{2}x^{2} + a_{1}x + a_{0}
=

a_{i} are any real numbers except a_{n}≠0.
The exponents are positive integers only.

a

Example:

n is the *degree* of the polynomial.

linear polynomial: a_{1}x + a_{0} AKA: mx+b

quadratic polynomial: a_{2}x^{2} + a_{1}x + a_{0}
AKA: ax^{2}+bx+c

cubic polynomial: a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}
AKA: ax^{3}+bx^{2}+cx+d

quartic: 4th power.

quintic: 5th power.

Domain of every polynomial function is R.

Range: "ends" of graph always go to ±∞

even degree, e.g. quadratic, quartic, sixth-degree: "half open"

If a_{n}<0 then ƒ first increases from -∞ to y_{max} then decreases to -∞.

i.e. both "ends" point down to -∞.

If a_{n}>0 then ƒ first decreases from +∞ to y_{min} then increases to +∞.

i.e. both "ends" point up to +∞.

odd degree, e.g. linear, cubic, quintic: R.

If a_{n}<0 then ƒ decreases from +∞ to -∞.

i.e. left "end" points up to +∞, right "end" points down to -∞

If a_{n}>0 then ƒ increases from -∞ to +∞.

i.e. left "end" points down to -∞, right "end" points up to +∞

Continuous everywhere (no gaps, holes, steps in graph).

Easy to differentiate: each term (a_{i}x^{i})'=ia_{i}x^{i-1}

and to integrate: each term ∫a_{i}x^{i}=(a_{i}/(i+1))x^{i+1} +c

Number of x-intercepts is ≤ degree. (Odd degree has at least one.)

Sum, difference, product and composition of two polynomial functions is polynomial.

Quotient of two polynomial functions p(x)/q(x) in general is not polynomial
unless q is a factor of p.
Usually a rational function.

Polynomial equation in standard form:

a_{n}x^{n} + a_{n-1}x^{n-1} +... + a_{2}x^{2} + a_{1}x + a_{0} = 0

A *root* is c such that ƒ(c)=0. i.e. c is a solution to this equation.

Has n roots/zeroes (some of which might be imaginary numbers,
which, if the polynomial has only real coefficients, are
complex conjugates (pairs of the form a+bi and a-bi)).

Formulas exist for solving linear, quadratic, cubic (Cardano) and quartic (Ferrari/Tartaglia)
polynomial equations,
but no higher (Abel).

Linear: x=-a_{0}/a_{1} =-b/m

Quadratic formula: x=(-b±√(b^{2}-4ac))/2a

Cubic and quartic too complicated...

(x-c) is a (linear) factor iff c is a zero, i.e. can divide P by c.

Fundamental theorem of algebra:

a polynomial of degree n has n linear
factors (x-c_{i}) which can be real, complex, repeated.

Repeated real roots are the touching the x-axis intercepts.

Rational roots theorem:

the only possible rational roots of a polynomial with integer coefficients
are of the form p/q where p is an integer factor
of a_{0} and q is an integer factor of a_{n}.

Example: a polynomial with integer coefficients.
Only the leading coefficient and the constant are needed:
6x^{n} + ... + 4 = 0

p=4, integer factors are ±{1, 2, 4}

q=6, integer factors are ±{1, 2, 3, 6}

All quotients p/q: ±{1,2,3,4,6,1/2,1/3,1/4,1/6,2/3,4/3}.
Any rational roots of this polynomial are in this set.

Descartes' rule of signs:

--number of positive roots is either the number of sign changes in P(x) or any multiple of 2 smaller.

--number of negative roots is either the number of sign changes in P(-x) or any multiple of 2 smaller.

Example:

P(x): 3x^{4}-5x^{3}-3x^{2}+x-6=0 has 3 sign changes, so it has either 3 or 1 positive root(s).

P(-x): 3x^{4}+5x^{3}-3x^{2}-x-6=0 has 1 sign change, so it has 1 negative root.