1. POLYNOMIALS

2. Polynomials A polynomial is a function of the form
where the
are real numbers and n is a nonnegative integer.
The domain of a polynomial function is the set of real numbers

3. The Degree of Polynomial Functions
The Degree, of a polynomial function in one variable is the largest power of x
Example
Below is a polynomial of degree 2
See Page 183 for a summary of the properties of polynomials of degree less than or equal to two

4. Properties of Polynomial Functions
The graph of a polynomial function is a smooth and continuous curve
A smooth curve is one that contains no Sharp corners or cusps
A polynomial function is continuous if its graph has no breaks, gaps or holes

5. Power Functions
A power function if degree n, is a function of the form
where a is a real number, and n > 0 is an integer
Examples (degree 4),
(degree 7) ,
(degree 1)

6. Graphs of even power functions
The polynomial function is even if n is even. The functions graphed above are even. Note as n gets larger the graph becomes flatter near the origin, between (-1, 1), but increases when x > 1 and when x < -1. As |x| gets bigger and bigger, the graph increases rapidly.

7. Properties of an even function
The domain of an even function is the set of real numbers
Even functions are symmetric with the
y-axis
The graph of an even function contains the points (0, 0) (1,1) (-1, 1)

8. Graphs of odd power functions
The polynomial function is odd if n is odd. The functions graphed above are odd. Note as n gets larger the graph becomes flatter near the origin,
-1 < x <1 but increases when x > 1 or decreases when
x < As |x| gets bigger and bigger, the graph increases for values of x greater than 1 and decreased rapidly for values of x less than or equal to -1.

9. Properties of an odd function
The domain of an odd function is the set of real numbers
Odd functions are symmetric with the
origin
The graph of an odd function contains the points (0, 0) (1,1) (-1,-1)

10. Graphs of Odd functions

11. Graphs of Even functions

12. Zeros of a polynomial function
A real number r is a real zero of the polynomial f (x) if f (r) =0
If r is a zero of the polynomial, then r is an x – intercept.
If r is a zero of the polynomial f (x) then
f (x) = (x – r) p (x), where p (x) is a polynomial

13. The intercepts of a polynomial
If r is an x – intercept of a polynomial x, then
f( r ) = 0
If r is an x – intercept then either
1. The graph crosses the x axis at r or
2. The graph touches the x axis at r