§1.3 Polynomial Functions The student will learn about: Polynomial function equations, graphs and roots.

Introduction to Polynomials A polynomial is an equation of the form f (x) = an xn + an-1 xn-1 + … + a1 x + a0 And graphs as a curve that wiggles back and forth across the x-axis.

Definition Def: A polynomial function is a function of the form f (x) = an xn + an-1 xn-1 + … + a1 x + a0 For n a nonnegative integer called the degree of the polynomial. The coefficients a0, a1, … , an are real numbers with an ≠ 0. The domain of a polynomial function is the set of real numbers.

Graphs of a Polynomials The shape of the graph of a polynomial function is connected to the degree and the sign of the leading coefficient an , and usually wiggles back and forth across the x-axis.

Graphs of a Polynomials Knowing the behavior of the ends of the graph helps us graph. What happens to y as x becomes very large or very small? That is, what happens at the tails? Polynomial: Tail Behavior Chart Leading coefficient positive Leading coefficient negative Degree even Both tails go up Both tails go down Degree odd Left down, right up Left up, right down

Graphs of a Polynomials Polynomial: Tail Behavior Chart Leading coefficient positive Leading coefficient negative Degree even Both tails go up Both tails go down Degree odd Left down, right up Left up, right down Remembering y = x 2 and y = x 3 may help!

Graphs of a Polynomials The graph of a polynomial function of positive degree n can cross the x-axis at most n times. An x-intercept is also called a zero or a root.

Polynomial Root Approximation. Theorem 1: If r is an x-intercept of the polynomial P (x) = an xn + an - 1 xn - 1 + an - 2 xn - 2 … + a1 x + a0 Let a n = the leading coefficient, and Let b = absolute value of the largest coefficient - Then |r| < (1 + | b | ) / | an | ) . This gives us the maximum and minimum values for roots and helps in our search.

Solving a Polynomial Solutions may be found algebraically. Let f (x) = 0 and solve for x : a. Factor – a bit of work that sometimes requires synthetic division (whatever that is). b. Use the quadratic formula when you have second degree factors. c. Use a graphing calculator and use the calc and zero buttons. Remember you are responsible for both algebraic and calculator methods.        I love my calculator!        9

Graphs of a Polynomials The graph of a polynomial has y-axis (even, vertical) symmetry if all of the exponents are even. y = x 4 – 2x 2 - 1 Note: both (x, y) and – x, y) are on the graph The graph of a polynomial has origin (point, odd) symmetry, if all of the exponents are odd. y = x 3 – 2x Note: both (x, y) and (- x, - y) are on the graph

Graphs of a Polynomials The graph of a polynomial function of positive degree n can have at most (n – 1) turning points. These points are called relative maximum and relative minimum points. A polynomial function is continuous with no holes or breaks.

Example y = x4 + 2 x3 – 4x2 - 8 x 1. x-intercepts I will factor using grouping x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2) = (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2) = x (x – 2)(x + 2)(x + 2) I will factor using grouping x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2) = (x 3 – 4x)(x + 2) I will factor using grouping x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2) I will factor using grouping x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2) = (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2) Hence the x-intercepts are 0, 2, and – 2 as a root of multiplicity two. Note that – 8 < r < 8. |r| < ( 1 + | b | ) / | an |

Example y = x4 + 2 x3 – 4x2 - 8 x 2. y-intercept Allowing x = 0 in the original equation gives a y = 0 which means that the y-intercept is the origin,

Example y = x4 + 2 x3 – 4x2 - 8 x 3. Tail behavior. The degree is even and the leading coefficient is positive so both ends go up.

Example y = x4 + 2 x3 – 4x2 - 8 x 3. Symmetry. The function is neither even nor odd so it has no symmetry.

Note: We will be finding the two relative minimum and the relative maximum with calculus although you already know how to do this with a graphing calculator!

Polynomial Function Review f (x) = an xn + an-1 xn-1 + … + a1 x + a0 The shape of the graph is connected to the degree and the leading coefficient. The graph wiggles back and forth across the x-axis. Finding the x-intercepts is important. Finding the relative minimums and maximums will become important.

Graphs of a Polynomials Graphing polynomials is difficult and time consuming. Calculus will aide us greatly in determining the x-intercepts, (the y-intercept is always easy!), and the maximum and minimum points of a polynomial. Although this is helpful in graphing it is really more helpful in life as these characteristics have many applications.

Summary. We had an introduction to polynomial functions and learned some of the properties of these functions. We had an introduction to rational functions. We learned about both the vertical and horizontal asymptotes associated with rational functions. We worked through an application that involved rational functions.

ASSIGNMENT §1.3; Page 13; 1 – 27 odd.