Polynomial FunctionsAFM Objective 1.01

Polynomials have more than one term:◦ 2x + 1◦ 3x2 + 2x + 1

The fancy explanation:◦ f(x) = anxn + an-1xn-1 + …+ a2x2 + a1x + a◦ Ex: 4x3 + 3x2 + 2x + 1

Simply put: the addition and subtraction of terms with different exponents going from largest to smallest

Polynomial Functions

Polynomial functions are named by their degrees

The largest exponent is the degree of the function

Polynomial Functions

4x3 + 3x2 + 2x + 1

This has a degree of 3

Leading Term:◦ The term with the highest exponent

◦ This function has a degree of 3

◦ By knowing the leading term we can analyze a function without graphing it

*The highest exponent may not be written first in the function!!

Polynomial Functions

4x3 + 3x2 + 2x + 1

What are the degrees of the following polynomial functions?

1. 3x5 – 2x4 + 9x – 2

2. -6x3 + 2x2 – 7x + 10

3. 2x11 + 5x6 – 3x2

4. 4x2 + 5x4 – 2x3 + x – 5

Quick Quiz 1

What are the degrees of the following polynomial functions?

1. 3x5 – 2x4 + 9x – 2

2. -6x3 + 2x2 – 7x + 10

3. 2x11 + 5x6 – 3x2

4. 4x2 + 5x4 – 2x3 + x – 5

Quick Quiz 1

Degree of 5

Degree of 3

Degree of 11

Degree of 4

Certain polynomial functions have similar qualities.

We will use the Leading Term to analyze the behavior of a function

For simplicity sake we will just use the function f(x) = axn

Polynomial Functions

If a is positive and n is an odd number what kind of equations will you get?◦ 2x◦ 4x3

◦ 7x5

Here’s an example:

axn

Here’s another example: Notice how the

straight line and this line both point down on the left and up on the right

The have the same “end behavior”

axn

All equations that have a positive a and odd n have the same end behavior

axn

If a is negative and n is odd it will change the end behavior (ex: -2x3)

axn

If a is positive and n is an even number what kind of equations will you get?◦ 2x2

◦ 4x4

◦ 7x6

Here’s an example: Notice that both

ends point the same way, up!

axn

All equations where a is positive and n is an even number will have the same end characteristics:

axn

If a is negative and n is an even number it will change the end behavior

Here’s an example: Notice that both

ends still point the same way but down

axn

All equations where a is negative and n is an even number the end behavior will be:

axn

What is the end behavior of the following functions?

1. f(x) = 3x3

2. f(x) = 4x6 – 2x2 + 1

3. f(x) = -3x2 + 2x – 5

4. f(x) = 4x – 5x5 + 3x3

Quick Quiz 2

What is the end behavior of the following functions?

1. f(x) = 3x3

2. f(x) = 4x6 – 2x2 + 1

3. f(x) = -3x2 + 2x – 5

4. f(x) = 4x – 5x5 + 3x3

Quick Quiz 2

Turning Points:◦ Where the graph changes direction

Turning Points

Turning Point

◦ It goes from increasing to decreasing or the other way around

This is a cubic (x3) function and it has 2 turning points

Turning Points

Turning Point

This is also a cubic (x3) function

Turning Points

Notice there are no Turning Points!

This function never changes it’s direction

*Cubic (x3) functions can have up to 2 Turning Points

This is a quartic (x4) function

Turning Points

Turning Point

This is also a quartic (x4) function

Turning Points

Notice there is just one Turning Point!

This function only changes direction once

*Quartic (x4) functions can have up to 3 Turning Points

If you look at the exponent on the Leading Term you can analyze up to how many Turning Points there can be

Turning Points

axn Has up to n – 1 turning points

Up to how many possible turning points do the following functions have?

1. 3x5 – 2x4 + 9x – 2

2. -6x3 + 2x2 – 7x + 10

3. 2x11 + 5x6 – 3x2

4. 4x2 + 5x4 – 2x3 + x – 5

Quick Quiz 3

Up to how many possible turning points do the following functions have?

1. 3x5 – 2x4 + 9x – 2

2. -6x3 + 2x2 – 7x + 10

3. 2x11 + 5x6 – 3x2

4. 4x2 + 5x4 – 2x3 + x – 5

Quick Quiz 3

4

2

10

3

A zero is the x value when the function crosses the x-axis

Zeros

Zeros This is a quartic

(x4) function Notice it has 4

zeros

This is also a quartic (x4) function

Zeros

Zeros Notice it has only

2 zeros The others are

“imaginary”*Quartic functions

can have up to 4 real zeros

If you look at the exponent on the Leading Term you can analyze up to how many Zeros there can be

Zeros

axn Has up to n real zeros

Remember that when a line crosses the x-axis, the y value is zero.

Any point that is (x, 0) is a “zero” or x-intercept

You can find some of the zeros by solving the equations

Zeros

Example: Determine all the real zeros for

This is a cubic function so it can have up to 3 real zeros

First set the function equal to zero (y is zero)

We need to factor to find the solutions

Zeros

x3 – 5x2 + 6x

Find a common factor and pull it out Factor what’s left inside the parenthesis

Zeros

x3 – 5x2 + 6x = 0x(x2 – 5x + 6) = 0x(x – 2)(x - 3) = 0

Set each part of the function equal to zero Solve each one for x

Zeros

x(x – 2)(x - 3) = 0x - 3 = 0x – 2 = 0x = 0

x = 0, 2, 3

Example 2: Determine all the real zeros for

We know this will have up to 4 real zeros Also, this equation is in quadratic form since

we can write it like:

Zeros

x4 – 8x2 + 15

(x2)2 – 8(x2) + 15

Two x2’s One x2 A #

In order to factor this properly we can replace the x2’s with the letter u

We now have an equation we can factor!

Zeros

(x2)2 – 8(x2) + 15(u)2 – 8(u) + 15

Factor u2 – 8u + 15

Remember, u is equal to x2

Plug x2 in for u

Zeros

(u)2 – 8(u) + 15(u – 3)(u - 5)

(x2 – 3)(x2 - 5)

Solve each one for x

Zeros

x2 – 5 = 0x2 – 3 = 0

x2 = 3 x2 = 5

x = ±

x = ±

Determine all the real zeros for

This can have up to 4 real zeros

Factor out the GCF first (always take the negative with the first number)

Zeros

-x4 – x3 + 2x2

-x2(x2 + x – 2)

Factor inside the parenthesis

Set everything equal to zero

Zeros

-x2(x2 + x – 2)-x2(x + 2)(x – 1)

Solve each part for x

Zeros

-x2 = 0

-x2(x + 2)(x – 1)

-1 -1x2 = 0x = ± 0

Notice that you have two answers of zero

This is called “Multiplicity”

Multiplicity- when you have a repeated zero (answer)

Zeros

x = ± 0

Solve each part for x

Zeros

x + 2 = 0 x – 1= 0

x = -2 x = 1