Polynomial functions and equations.

ƒ(x) = anxn + an-1xn-1 +... + a2x2 + a1x + a0 =    
ai are any real numbers except an≠0. The exponents are positive integers only.


n is the degree of the polynomial.
linear polynomial: a1x + a0    AKA: mx+b
quadratic polynomial: a2x2 + a1x + a0    AKA: ax2+bx+c
cubic polynomial: a3x3 + a2x2 + a1x + a0    AKA: ax3+bx2+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 an<0 then ƒ first increases from -∞ to ymax then decreases to -∞.
      i.e. both "ends" point down to -∞.
    If an>0 then ƒ first decreases from +∞ to ymin then increases to +∞.
      i.e. both "ends" point up to +∞.
  odd degree, e.g. linear, cubic, quintic: R.
    If an<0 then ƒ decreases from +∞ to -∞.
      i.e. left "end" points up to +∞, right "end" points down to -∞
    If an>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 (aixi)'=iaixi-1
and to integrate: each term ∫aixi=(ai/(i+1))xi+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:
anxn + an-1xn-1 +... + a2x2 + a1x + a0 = 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=-a0/a1 =-b/m
  Quadratic formula: x=(-b±√(b2-4ac))/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-ci) 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 a0 and q is an integer factor of an.
Example: a polynomial with integer coefficients. Only the leading coefficient and the constant are needed: 6xn + ... + 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.
P(x): 3x4-5x3-3x2+x-6=0 has 3 sign changes, so it has either 3 or 1 positive root(s).
P(-x): 3x4+5x3-3x2-x-6=0 has 1 sign change, so it has 1 negative root.