Properties of Polynomial Functions:
Given that f(x) and g(x) are polynomials with real coefficients,
the following are true:
Poly1  Division Algorithm: If polynomial
f(x) divided by polynomial D(x) results in quotient Q(x) with with
remainder R(x), then we may write
f(x) = D(x)●Q(x) + R(x).
Note that this is the same result that applies to zero degree
polynomials, i.e. real and complex numbers. For example, if 42÷5
= 8 with remainder 2, then 42 = 5●8 + 2. We used the division
result to write the dividend 42 in terms of quotient and remainder.
Note: Dividing both sides of f(x) = D(x)●Q(x) + R(x) by Q(x) gives
us f(x)/D(x) = Q(x) + R(x)/D(X), the result of f(x) divided by a
divisor polynomial D(X). So in the previous example, we write the
result of division as 42÷5 = 8 + 2/5.
Note: This Division Algorithm, applied to polynomials, implies that
we can divide out polynomial f(x) by D(x) uses the same division
method used for real numbers. Divide leading term of f(x) by
leading term of D(x) to obtain a quotient. Multiply the
quotient by the divisor D(x). Subtract the product from like
terms of f(x). Carry Down the next term. Repeat this
division process until the degree of the last leading term is less
than that of the leading term of the divisor D(x). The remainder is
placed over the divisor as a fractional part of the answer.
Note: The shortcut process Synthetic Division, used to
divide f(x) by (xc), is nothing more than shorthand for polynomial
division. Justification is left as an exercise.
Poly2  Remainder Theorem: If f(x) is divided
by (x  c) with remainder r, then f(c) = r.
Poly3  Factor Theorem: f(x) divided by
g(x) results in h(x) with zero remainder if and only if g(x) is a
factor of f(x).
Poly4  Properties of a Polynomial Zero: x = a
is a zero of a polynomial f(x) if f(a) = 0. Furthermore, if x=a is a
zero, then
 (x  a) is a factor of f(x) and division by (xa) results in
remainder = 0.
Also f(x) may be written as f(x) = (x  a)●Q(x) where Q(x) is the
result of dividing f(x) by Q(x).
 (a, 0) is an xintercept of the graph of f(x) if "a" is a real
zero,
Poly5  Leading Terms of Polynomial Function Graphs:
If f(x) is a polynomial, it's leading term will determine the behavior
of the graph on the far right and far left. e.g. If the leading term
is positive for positive values of x, then the graph will rise on the
far right. If the leading term is positive for negative values of x,
then the graph will rise on the far left. This is commonly known as
The Leading Term Test or The Leading Coefficient Test.
Poly6  Multiplicity Properties of Zeros:
A zero x=c of a polynomial f(x) has even multiplicity if and only if
it's related factor in the factorization of f(x) is (x  c)^{N}
where N is even. A zero x=c of a polynomial f(x) has odd
multiplicity if and only if it's related factor in the factorization
of f(x) is (x  c)^{N} where N is odd.
If the multiplicity is even, the graph will "bounce off" the
xintercept. i.e. f(x) will be positive both left and right of
the intercept or negative both right and left of the intercept.
If the multiplicity is odd, the graph will "pass through" the
xintercept. i.e. f(x) will be positive on one side of the
intercept and negative on the other.
Poly7  Intermediate Value Theorem: If f(x) is
a polynomial, and f(a) ≠ f(b) for a<b,
then f(x) takes on ever value from f(a) to f(b) in the closed interval
[a,b].
Applied to polynomial zeros, The Intermediate Value Theorem states
that if f(a) < 0 and f(b) > 0, then there must be a value x=c in the
interval [a,b] such that f(c) = 0.
Poly8  Fundamental Theorem of Algebra: If f(x)
is a polynomial with degree n, then there is at least 1 complex zero
x=c. Furthermore, if f(x) has degree n ≥ 1 with nonzero leading
coefficient a_{n}, then
f(x) has exactly n linear factors and may be written as f(x)= a_{n}(x
 c_{1})(x  c_{2}) . . . (x  c_{n}) where c_{1},
c_{2}, c_{3}, ....,c_{n} are real or complex
zeros and some of the zeros and associated factors may be repeated.
The power on any repeated factor is known as its multiplicity. Factors
that are not repeated have multiplicity = 1.
Poly9  Descartes' Rule of Signs: If f(x) is a
polynomial with real coefficients with terms listed from highest to
lowest power with k sign changes from term to term, then there will be
"k" positive real zeros or "k  m" real zeros where m is some even
integer.
For example, if f(x) = x^{4}  3x^{3} + 2x^{2}
+ x  1, there are k=3 sign changes so there will be k=3 positive real
zeros or k  m = 3  2 = 1 positive real zero.
If f(x) is a polynomial with real coefficients with terms listed
from highest to lowest power with k sign changes from term to term of
f(x), then there will be "k" negative real zeros or "k  m" real
zeros where m is some even integer.
For example, if f(x) = x^{4}  3x^{3} + 2x^{2}
+ x  1, then f(x) = x^{4} + 3x^{3} + 2x^{2}
 x  1 there are k=1 sign changes so there will be k=1 negative real
zeros or k  m = 1  0 = 1 negative real zero. Here the only
even integer less than 1 is m=0.
Note: Missing terms (coefficient = 0) are not considered. For
example, f(x) = x^{5}  3x^{2}  2x^{2}
 1 would have zero sign changes.
Poly10  Upper Bounds of Real Zeros: Let f(x)
be a polynomial with real coefficients and leading coefficient a
that is divided by (x  c) using synthetic division, where c
> 0 and c is real.
c is an upper bound for all real zeros of f(x) if the
leading coefficient a is positive provided none of the numbers
in the bottom row of the synthetic division are negative. An example
is shown below:
c is an upper bound for all real zeros of f(x) if c >
0 and leading coefficient a is negative provided none of the
numbers in the bottom row of the synthetic division are positive. An
example is shown below:
Poly11  Lower Bounds of Real Zeros: Let f(x)
be a polynomial with real coefficients that is divided by (x  c)
using synthetic division, where c < 0 and c is real.
c is a lower bound for all real zeros of f(x) if the
bottom row of the synthetic division alternates in sign. If a value in
the bottom row is zero, it can be considered to be positive or
negative as needed to show the alternating pattern. An example
is shown below:
Poly12  Complex Zeros Occur In Conjugate Pairs:
If f(x) is a polynomial with real coefficients and has one complex
zero x = a + bi, then x = a  bi will also be a zero. Furthermore, x^{2}
 2ax + a^{2} + b^{2} will be a factor of f(x).
