Polynomials of Higher Degree2-2

Polynomials and Their Graphs

Polynomials will always be continuous

Polynomials will always have smooth turns.

End Behavior(leading coefficient test

Odd Power Positive leading coefficient

Negative leading coefficient

Even Degree Positive leading coefficient

Negative leading coefficient

Zeros

If f(x) is a polynomial, the following statements are equivalent X = a is a zero of the function X = a is a solution to the equation f(x)=0 (x-a) is a factor of the polynomial f(x) a is an x-intercept of the graph of f(x)

Finding zeros

Find all real zeros of

Analyzing a polynomial

Find all real zeros of

Use your calculator to find all relative Extrema (min/max)

Finding the Zeros of a Polynomial

Graph the function on your calculator, then use it to find the roots.

10

8

6

4

2

-2

-4

-6

-8

-10

-12

-15 -10 -5 5 10 15 20 25

Repeated Root

If a root or factor is repeated it behaves differently on the graph.

If is a root and k is even it touches the axis and does not cross

If is a root and k is odd it flattens out and crosses the axis

Repeated Roots

K is even ex.

3

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

-3

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Repeated Roots

K is odd ex. 3

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

-3

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Pg 108 # 17-33, 35, 41, 43