Polynomial Theorems

Fixed Height

A function f is constant on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) = f(x2)

Decreases from Left to Right

A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2)

Increases from Left to Right

A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2)

Odd multiplicity

Cross through

Global (Absolute) Maximum

Highest point out of the entire graph

The Complex Conjugates Theorem

If a + bi is a zero of a polynomial with real coefficients, then a - bi must also be a zero

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, with n>0, then the equation f(x)=0, has exactly n complex solutions as long as repeated solutions are counted separately.

Global (Absolute) Minimum

Lowest point out of the entire graph

Descartes's Rule of Signs (Part 3)

The number of possible imaginary solutions is the degree of the poly. funct. minus the min. number of negative solutions and the minimum number of positive solutions

Descartes's Rule of Signs (Part 2)

The possible number of negative real zeros is equal to the number of times of f(-x) changes signs or less than that by an even number

Descartes's Rule of Signs (Part 1)

The possible number of positive real zeros of a polynomial function is equal to the number of times f(x) changes signs or less than that by an even number.

Even multiplicity

Touch

Local Maximum (Relative-close)

extrema; y-coordinate of turning point if the point is higher than all nearby points

Local Minimum (Relative)

extrema; y-coordinate of turning point if the point is lower than all nearby points

Multiplicity

if (x-r)^m is a factor of a polynomial (f) & (x-r)^m+1 isn't a factor of f, then r is called a zero of multiplicity m of f

Corollary

if f(x) is a poly. of degree n where n > 0, then the equation f(x)=0 has n solutions provided each solution repeated twice is counted as 2 solutions

Irrational Conjugates Theorem

suppose f is a poly. funct. with rational coefficients, and a+ b is a rational number such that √b is irrational; if a + √b is a zero of f, then a - √b is a zero of f

Theorem of Turning Points

the graph of every poly. function of degree n has at most n-1 turning points; if a poly. function has n distinct real zeros, then the graph has exactly n-1 turning points