Polynomial and Rational Functions - Lecture Notes | MAT 150, Study notes of Trigonometry

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Polynomial and Rational Functions

4.Polynomial Functions of Higher Degree

October 6, 2009

Definition: Polynomial Function

Let n be a nonnegative integer, and let an, an− 1 , ..., a 2 , a 1 , a 0 be real numbers with an 6 = 0. The function

f (x) = anxn^ + an− 1 xn−^1 + ... + a 2 x^2 + a 1 x + a 0

is called a polynomial function of x with degree n. The coefficient an is called the leading coefficient, and a 0 is the constant.

Example 1

For each of the given functions, determine whether the function is a polynomial functions. If it is a polynomial function, state the degree of the polynomial.

a. f (x) = 3 − 2 x^5 b. F (x) =

x + 1 c. g (x) = 2 d. h(x) = 3x^2 − 2 x + 5 e. H(x) = 4x^5 (2x − 3)^2 f. G (x) = 2x^4 − 5 x^3 − 4 x−^2

Definition: Power Function

Let n be a positive integer and the coefficient a 6 = 0 be a real number. The function f (x) = axn

is called a power function of degree n.

Power functions with even powers look similar to the square function.

Power functions with odd powers (other than n = 1) look similar to the cube function.

Real Zeros of Polynomial Functions

If f (x) is a polynomial function and a is a real number, then the following statements are equivalent.

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

Consider the polynomial function f (x) = x^2 − 1.

Multiplicity of a zero and relation to the graph of a polynomial

If a is zero of f (x), then:

Multiplicity f (x) on either Graph of Function of a side of x = a at the Intercept Even Does not change sign Touches the x-axis (turns around) at point (a, 0) Odd Changes sign Crosses the x-axis at point (a, 0)

End Behavior

As x gets large in the positive (x → ∞) and negative (x → −∞) directions, the graph of the polynomial

f (x) = anxn^ + an− 1 xn−^1 + ... + a 2 x^2 + a 1 x + a 0

has the same behavior as the power function

y = anxn.