Welcome to MM250 Unit 6 Seminar: Polynomial Functions To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here

Polynomials are functions that consist of whole number powers of x, like f(x) = 5x 4 + 3x 3 - x 2 + x - 7 The graphs of polynomial functions are smooth curves with turning points.

f(x) = x 3 + 4x 2 - x - 5

f(x) = x 4 - 5x 2 + 4

f(x) = -2x 4 + 5x 3 - 2x 2 + x + 1

End Behavior As | x | ----> ∞ the largest power term dominates Ex: f(x) = x 3 + 4x 2 - x - 5 This term determines the end behavior of the graph.

End Behavior General polynomial: Leading term (highest power term) is Ex: f(x) = 6x 5 + 2x leading term is 6x 5

End Behavior Suppose n is an even number: Then x n is always …… for both + and - x's If n is an odd number: Then x n is ….. for + x's and ….. for - x's

End Behavior Example: Leading term is 3x 4 Example: Leading term is -3x 4

End Behavior Example: Leading term is 8x 3 Example: Leading term is -7x 3

Leading Coefficient Test If n is even: if a n is > 0, rises left and right if a n is < 0 falls left and right If n is odd: if a n is > 0, falls left and rises right if a n is < 0 rises left and falls right

Zeros of a function where f(x) = 0

Intermediate Value Theorem If f( x 1 ) is > 0 and f( x 2 ) is < 0 Then the function has a zero between x 1 and x 2

Intermediate Value Theorem f(x) = 2x 4 - 4x Is there a zero between x = -1 and x = 0?

Finding Zeros Ex: f(x) = x 2 - 5x + 6 Find where f(x) = 0, x 2 - 5x + 6 = 0 Factor: (x - 3)(x - 2) = 0 x - 3 = 0 or x - 2 = 0 x = 3 or x = 2 Notice: Each factor divides evenly into f(x)

Finding Zeros When you have a more complex polynomial, like f(x) = x 5 - 4x 3 + 2x - 6, can you factor it like you did with the quadratic equation? It turns out that you can. Any polynomial f(x) can be factored like f(x) = a(x - ?)(x - ?)(x - ?)... (x - ?) where the a and the ?'s are numbers. However, the ?'s may not be real numbers. We'll talk about that later. But notice that each factor (x - ?) divides evenly into the polynomial. So we want to be able to divide a polynomial by a binomial. We do this by long division.

Page 331 #5 Long Division

Page 331 #25 Synthetic Division

Roots "zeros of f(x)" same as "roots of f(x) = 0" They may be real or complex numbers. If they are real, they may be rational (can be written as fractions)

Rational Roots Theorem Ex: f(x) = x 4 - x 3 + x 2 - 3x - 6 Theorem says that any possible rational root will be of the form: (+ or - )(factor of constant term)/(factor of coefficient of highest order term) Factors of 6 are: 1, 2, 3, 6 Factors of 1 are: 1 Possible rational roots are positive or negative 1/1, 2/1, 3/1, 6/1 So 1, 2, 3, 6, -1, -2, -3, -6

Rational Roots Theorem Ex: f(x) = x 4 - x 3 + x 2 - 3x - 6 Possible rational roots are 1, 2, 3, 6, -1, -2, -3, -6 These are POSSIBLE rational roots. Some or all may not actually be roots. To determine which of them are, plug each into the function and see if you get 0. f(1) = -8 not a root f(-1) = 0 is a root Plug them all in you get that -1 and 2 are roots.

Example of Finding roots f(x) = 2x 3 + 6x 2 + 5x + 2

Example of Finding roots

ax 2 + bx + c = 0 Quadratic Formula gives solutions:

2x 2 + 2x + 1 = 0 Quadratic Formula gives solutions:

Example of Finding roots f(x) = 2x 3 + 6x 2 + 5x + 2 Roots are: -2, (-1 + i)/2 and (-1 - i)/2 3 roots for a 3rd degree polynomial