A polynomial function is a function that can be written as f(x)=axn+bxn-1+cxn-2...gx+h, where a, b, c, ..., g, and h are real numbers and n is a positive integer. The key properties of a polynomial function are its degree, leading coefficient, factors, zeros, and relationship between factors and x-intercepts. The graph of a polynomial cannot intersect the x-axis more times than the number of distinct factors it has.

2. A polynomial function is a function
that can be written as:
 f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h
Where :
 a, b, c, …, g, and h are real numbers
 n is a positive integer.

3. POLYNOMIALS NOT POLYNOMIALS
 f(x)=½x²+2x-3  F(x)= ίxˉ²
 Why?  Why not?
 Correct form  ί is not a real number
 ½ , 2, and 3 are real  -2 isn’t a positive number
numbers
 2 and 1 are both positive
integers.

4.  Standard form:
 f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h
 Everything in the equation is
multiplied together.
 Factored form:
 f(x)=(x-r1)(x-r2)(x-r3)…

5.  Degree: the highest power of “x” in standard
form.
 Ex. : f(x)=½x²+2x-3 Degree = 2
 Leading coefficient: coefficient of the first
term if the terms are in descending order.
 f(x)=½x²+2x-3 Leading Coefficient = ½

6.  The factors of a polynomial are the
quantities of the function when in
factored form
 Ex: f(x)=(x-5)(x+7)(x-2)
 There are three factors in this
polynomial. They are x-5, x+7, and x-2.

7.  The zeros of a function are the numbers which
can be inserted into “x” so that the function
equals zero.
 Example:
f(x)=(x-4)(x+7) f(x)=(x-4)(x+7)
f(4)=(4-4)(4+7) f(-7)=(-7-4)(-7+7)
f(4)=(0)(11) f(-7)=(-11)(0)
f(4)=0 f(-7)=0
 The zeros of this function are 4 and -7.
 When graphed, the zeros are the x-intercepts.

8.  The number of zeros in a function are the same
as the intercepts. Therefore, an equation with
three different factors, such as
f(x)=(x-2)(x+3)(x-7), would have three x-
intercepts at (2,0), (-3,0), and (7,0).
 When two of the factors in an function are the
same (ex.: f(x)=(x-2)(x+3)(x-2)), then the graph
will, in this case, have two x-intercepts instead
of three, with the curved line sitting on the x-
axis instead of crossing it and intersecting in
three places.

9.  The graph of a function cannot intersect the x
axis any more time than the number of factors
it ha.
 When the factors of a function with just two
factors are the same (ex.: f(x)=(x+3)(x+3)), the
function has a “multiplicity of two.”