POLYNOMIALS

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

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

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

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)

Graphs of even power functions The polynomial function is even if n 2 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.

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)

Graphs of odd power functions The polynomial function is odd if n 3 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 < -1 . 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.

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)

Graphs of Odd functions

Graphs of Even functions

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

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