Higher degree . A degree 0 ( ) is a function of the form

A degree 1 polynomial () is a function of the form

A degree 2 polynomial () is a function of the form

A degree 3 polynomial () is a function of the form

A degree 4 polynomial is a function of the form

A degree d polynomial is a function of the form

Which of these are polynomials? What are their degrees?

5 1/x

x + 1 4x5 − x4 + 3x + 2 x − 1

(x + 3) · (x + 2) · (x + 1).

Degree d polynomials coming from modelling You have a 3 × 2 sheet of cardboard. Make a square cut at each corner. Fold the resulting flaps up to make an open topped box. What is the volume of this box in terms of the size of each cut?

x x

1 2

Graphs of polynomials: Sketch the graphs of some degree three polynomials, How many turning points does each one have? How many x-intercepts?

x3 x(2x2 + 1) 3x(x − 1)(x + 1) x(x − 1)2 Number of roots Number of turning points Zoom way out on these graphs let x go from -1,000 to +1,000. Sketch these graphs again

x3 x(2x2 + 1) 3x(x − 1)(x + 1) x(x − 1)2 How do they compare?

These statements are generally true: Facts

(1) A degree d polynomial has at most zero’s.

(2) A degree d polynomial has at most turning points.

d 1 (3) The graph of the degree d polynomial adx + . . . a1x + a0 zoomed out sufficiently looks like the graph of 3

What is the least degree a polynomial could have if this is its graph?

Back to the graph of x(x − 1)2 on the previous page. It has a root at and at . The roots look different though, right?

Close to each of these zeros which power function does the graph look like? Maybe even give local sketches.

If a polynomial is factored, you can read off its graph: If a polynomial factors as p(x) = q(x) · (x − c)p with q(c) 6= 0 then close to x = c the graph of p(x) looks like: 4

If a polynomial is factored, you can read off its graph: Example: To graph y = x · (x + 1)2 · (x − 2)3 (1) Find the zeros and indicate them on the graph (don’t worry too much about scale). (2) Close to each of these zero’s the function is basically a shifted/scaled/flipped power function. • Near x = 0: y ∼ x · (0 + 1)2 · (0 − 2)3 =

• Near x = −1: y ∼ −1 · (x + 1)2 · (−1 − 2)3 =

• Near x = 2: y ∼ 2 · (2 + 1)2 · (x − 2)3 =

(3) Interpolate smoothly between. (4) For big x-values, the function is basically y ∼

How does this compare with a computer generated graph? Your turn: To graph y = x2 · (x − 2)4 · (x + 1) (1) Find the zeros and indicate them on the graph (don’t worry too much about scale). (2) Close to each of these zero’s the function is basically a shifted/scaled/flipped power function. • Near x = 0: y ∼

• Near x = −1: y ∼

• Near x = 2: y ∼

(3) Interpolate smoothly between. (4) For big x-values, y ∼

How does this compare with a computer generated graph?