Welcome to MM250

Unit 6 Seminar:

Polynomial Functions

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Polynomials are functions that consist of whole number

powers of x, like

 

f(x) = 5x4 + 3x3 - x2 + x - 7

The graphs of polynomial functions are smooth curves with

turning points.

f(x) = x3 + 4x2 - x - 5

f(x) = x4 - 5x2 + 4

f(x) = -2x4 + 5x3 - 2x2 + x + 1

End Behavior

As | x | ----> ∞

the largest power term dominates

Ex: f(x) = x3 + 4x2 - x - 5

This term determines the end behavior of the graph.

End Behavior

General polynomial:

Leading term (highest power term) is

Ex: f(x) = 6x5 + 2x3 - 5 leading term is 6x5

End Behavior

Suppose n is an even number:

Then xn is always …… for both + and - x's

If n is an odd number:

Then xn is ….. for + x's and

….. for - x's

End BehaviorExample: Leading term is 3x4

Example: Leading term is -3x4

End BehaviorExample: Leading term is 8x3

Example: Leading term is -7x3

Leading Coefficient Test

If n is even: if an is > 0, rises left and right

if an is < 0 falls left and right

If n is odd: if an is > 0, falls left and rises right

if an is < 0 rises left and falls right

Zeros of a function

where f(x) = 0

Intermediate Value Theorem

If f( x1 ) is > 0 and f( x2 ) is < 0

Then the function has a zero between x1 and x2

Intermediate Value Theorem

f(x) = 2x4 - 4x2 + 1

Is there a zero between x = -1 and x = 0?

Finding Zeros

Ex: f(x) = x2 - 5x + 6

Find where f(x) = 0, x2 - 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 ZerosWhen you have a more complex polynomial,

like f(x) = x5 - 4x3 + 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) = x4 - x3 + x2 - 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) = x4 - x3 + x2 - 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) = 2x3 + 6x2 + 5x + 2

Example of Finding roots

Example of Finding roots

ax2 + bx + c = 0

Quadratic Formula gives solutions:

a

acbbx

2

42

2x2 + 2x + 1 = 0

Quadratic Formula gives solutions:

4

42 x

4

42 x

4

22 i

2

1 i

Example of Finding roots

f(x) = 2x3 + 6x2 + 5x + 2

Roots are: -2, (-1 + i)/2 and (-1 - i)/2

3 roots for a 3rd degree polynomial