DrDelMath

SUMMARY
Chapter 3:  Polynomial Functions

Section 3.1: Quadratic Functions

Everything is contained in the Special Topic titled Quadratic Functions listed under Suplemental Material
Samples of the graphs of quadratic functions are shown in Graphs of Quadratic Functions

Section 3.2: Polynomial Functions of Higher Degree

Definition:  A polynomial function is a function whose rule may be written in the form
    
where each a_i is a real number and n is a natural number.

Definition:  The exponent n of the leading term a_nxⁿ of the polynomial function f, is called the degree of the function f.

Fact:  For domain elements far from the origin, the leading term in a polynomial function dominates the entire expression when calculating range elements.

Fact:  The graph of a polynomial function is a continuous smooth graph with no sharp corners.

Fact:  The graph of a polynomial function f of degree n can have no more than n x-intercepts. The graph “tries” to have exactly n x-intercepts.

Fact:  The graph of a polynomial function f of degree n can have no more than n - 1 turning points (humps). The graph “tries” to have exactly n-1 humps.

Equavalent Statements Related to Real Zeros of Polynomial Functions
If f is a polynomial function whose rule is given by then the following statements are equivalent.

1. k is a real zero of the function f.
2. k is a solution of the polynomial equation = 0.
3. x - k is a factor of the polynomial .
4. (k, 0) is an x-intercept of the graph of the function f.
   

Definition: If f is a function whose rule is given by and if (x-a)^k is a factor of  , then a is a zero of multiplicity k of the function f.

Fact:  If a is a zero of multiplicity k and k is an odd number, then the graph of f crosses the x-axis at (a, 0).

Fact:  If a is a zero of multiplicity k and k is an even number, then the graph of f touches but does not cross the x-axis at (a, 0).

Intermediate Value Theorem
Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a)f(b) then, in the interval [a, b]  f takes on every value between f(a) and f(b).

The Intermediate Value Theorem has many applications, but the one of most interest in College Algebra is the following.
Application:  If f is a polynomial function such that f(a) < 0 and f(b) > 0, then f has an x-intercept (a real zero) between a and b.

Section 3.3: The Division Algorithm for Polynomials

Division Algorithm for Natural Numbers:  If a and b are natural numbers then there are unique natural numbers q and r such that

Definition:  If the remainder r in the division algorithm is 0, then we say that a is divisible by b.

Process: Long Division is a process for calculating the quotient q and remainder r whose existence is assured by the Division Algorithm. Long Division is also used to calculate the decimal equivalent of a fraction.

Division Algorithm for Polynomials: If p and d are polynomials with real coefficients, then there are unique polynomials q and r with real coefficients such that

Definition:  If the remainder r in the division algorithm is 0, then we say that the polynomial p is divisible by the polynomial d.

Process: Long Division and Synthetic Division are processes for calculating the quotient q and remainder r whose existence is assured by the Division Algorithm.

Remainder Theorem:  If a polynomial function f is divided by the polynomial function whose rule is g(x) = x - k, then the remainder is f(k).

Factor Theorem:   If g is a polynomial function whose rule is g(x) = x - k then g is a factor of a polynomial function f if and only if f(k) = 0.

Fact:  The above statement means:

• If g(x) = x – k and g is a factor of the polynomial function f, then f(k) = 0.
• If f(k) = 0 for a polynomial function f, then the function defined by g(x) = x – k is a factor of f.

Fact:  The complex number k is a zero of the polynomial function f if and only if the function whose rule is g(x) = x - k is a factor of f.

Section 3.4: Zeros of Polynomial Functions

The Fundamental Theorem of Algebra:
If f is a polynomial function of degree n > 0, then f has at least one complex zero.

Fact:  If f is a polynomial function of degree n > 0 then f has exactly n complex zeros.

Fact:  Suppose f is a polynomial function with real coefficients.  If the complex number a + bi is a zero of the function f, then its conjugate a - bi is also a zero of the function f.
This fact is usually remembered by observing that complex zeros of polynomial functions occur in conjugate pairs.

Fact:  Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no linear factors with real coefficients.

Observation: Consider the set of all polynomials together with the usual operations of addition and multiplication. In that set linear polynomials with real coefficients are prime polynomials. In that set quadratic polynomials with no linear factors with real coefficients are prime polynomials.

Fact:  In view of this observation the previous fact may be restated as follows:
Every polynomial of degree n > 0 with real coefficients can be written as the product of prime polynomials.

Possible Rational Zeros:  If f is a polynomial function f whose rule is with integer coefficients, then every rational zero has the form  such that:

• the only common factor of p and q is 1
• p is a factor of the constant term a₀
• q is a factor of the leading term a_n
  

Section 3.5: Mathematical Modeling