List the degree number and parity of the following types of polynomial functions and write an example equation for each.

Type of polynomial / Degree number / Degree Parity / Examples
Constant
Linear
Quadratic
Cubic
Quartic
Quintic

Graphs of Polynomials

Polynomials are ______and ______everywhere.

  • A continuous function means that it can be drawn ______.
  • There are no jumps or holes or vertical asymptotes in the graph of a polynomial function.
  • A smooth curve means that there are no sharp turns (like an absolute value) in the graph of the function.

Leading Coefficient Test (right hand behavior)

  • If the leading coefficient of the polynomial is positive, then the right hand side of the graph will

______towards ______.

  • If the leading coefficient of the polynomial is negative, then the right hand side of the graph will

______towards ______.

Degree of the Polynomial

  • If the degree of the polynomial is even, both ends will ______. If the degree of the polynomial is odd,

the ends will ______.

This even-same, odd-changes notion occurs in many mathematical areas.

Zeros of a Polynomial Function

  • An 8th degree polynomial in one variable has at most ____ real zeros.
  • The degree is the same number of complex zeros (which is the total of all zeros both ______and ______) (Fundamental Theorem of Algebra).
  • If a 6th degree polynomial has 4 real zeros, then it will have ___ imaginary zeros.
  • If a function has imaginary zeros they will always occur in ______.

So if 7i is a zero then ______is also a zero.

  • A 4th degree polynomial in one variable has at most ____ turns(max or min extrema).
  • The rule is the maximum number of turns is always ______than the degree. Think of the quadratic function which is degree _____. It always has ____ turn.
  • Some of the roots may be repeated. These are called repeated roots. Repeated roots are tied to a concept called ______.
  • If the multiplicity is odd (such as occurring once or 3 times), the graph will ______the x-axis at that zero.
  • If the multiplicity is ______, the graph will touch or kiss the x-axis at that zero. That is, it will bounce back and stay on the same side of the axis.

Sketch a quintic polynomial (degree ____) with max turns and a positive leading coefficient that has a double root (multiplicity of 2) at the origin. It also has real zeros at -5, -1, and 3

.

Write an example equation of least degree in factored form forthe quintic graph above.

Write least degree equations for each of the following (leave in factored form)

What is the degree? How many real zeros? How many imaginary?

Given the following zeros:

  1. { 8, 0, -7 multiplicity of 2}
  1. { -1, 0, 3}
  1. { -i, 2, 9}
  1. {6i,-10, 0, 5 multiplicity of 2}