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 (x-c), 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 (x-a) 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 x-intercept 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 x-intercept.  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 x-intercept.  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 non-zero leading coefficient a_n, then

f(x) has exactly n linear factors and may be written as f(x)= a_n(x - c₁)(x - c₂) . . . (x - c_n) where c₁, c₂, c₃, ....,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) = x⁴ - 3x³ + 2x² + 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) = x⁴ - 3x³ + 2x² + x - 1,  then f(-x) = x⁴ + 3x³ + 2x² - 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) = -x⁵ - 3x² - 2x²  - 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, x² - 2ax + a² + b² will be a factor of f(x).