Opener

Use the resources on the internet and each other to help you relearn the concept.

Use whiteboards.Use laptops.Work together.

Opener - Solutions How well were you able to answer the question?

We will be reviewing our previous assignment next. So have your questions ready.

2.5 Writing a Polynomial Function SWBAT write a polynomial function

given the real and complex zeros.

Write down the date, Section #, Title and Learning Objective(s) into your notebook.

Key ConceptsLOOK LISTEN

Write down what you want into “Key Concepts” as you Look & Listen.

Complex Number Factoids:

1 i 2 1i

2 2( )( )a bi a bi a b

Difference of Squares:2 2( )( )a b a b a b

2 2x a b x a b x a b

Key ConceptsLOOK LISTEN

Write down what you want into “Key Concepts” as you Look & Listen.

Linear Factorization Theorem

If f(x) is a polynomial of degree n where n>0, f has precisely n linear factors.

1 2( ) n nf x a x c x c x c K

Where c1, c2, … cn are complex numbers.

Key ConceptsLOOK LISTEN

Write down what you want into “Key Concepts” as you Look & Listen.

Linear Factorization Theorem Example

2( ) 6 9f x x x

• This is a second degree polynomial and therefore must have precisely 2 complex zeros.

( ) ( 3)( 3)f x x x • After factoring, we can see that the zero is 3 and is

repeated twice. i.e. it has a multiplicity of 2.How would the graph intersect at this zero? How do you know?

Key ConceptsLOOK LISTEN

Write down what you want into “Key Concepts” as you Look & Listen.

Linear Factorization Theorem Example

3( ) 4f x x x

• This is a second degree polynomial and therefore must have precisely 3 complex zeros.

( ) ( 2 )( 2 )f x x x i x i • After factoring, we can see that the zeros are

{0, +2i, -2i}. How many intersection points would the graph have on the x-axis? Why?

Key ConceptsLOOK LISTEN

Write down what you want into “Key Concepts” as you Look & Listen.

If I am given the zeros, how do I find the polynomial?

Take each zero and replace in the “Linear Factorization Theorem.”

1 2( ) nf x x xc cc x K

How do we get there?

SWBAT write a polynomial function given the real & complex zeros.

LOOK LISTEN

Find a 2nd degree polynomial that has the given zeros of: {-4, 7}.

Now let’s try together.Use whiteboards.

SWBAT write a polynomial function given the real & complex zeros.

Find a 3rd degree polynomial that has the given zeros of: {0, 1, -2}.

Now let’s try together.Use whiteboards.

SWBAT write a polynomial function given the real & complex zeros.

Find a 2nd degree polynomial that has the given zeros of: {-5 with multiplicity of 2}.

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 3rd degree polynomial that has the given zeros of: {0 with multiplicity of 2, -1}.

How do we get there?LOOK LISTEN

SWBAT write a polynomial function given the real & complex zeros.

Find a 2nd degree polynomial that has the given zeros of: .{ }i

* Complex numbers ALWAYS come in pairs. i.e. a+bi and a-bi.

Now let’s try together.Use whiteboards.

SWBAT write a polynomial function given the real & complex zeros.

Find a 4th degree polynomial that has the given zeros of: .{ 1,2, 3 }i

* Complex numbers ALWAYS come in pairs. i.e. a+bi and a-bi.

Now let’s try together.Use whiteboards.

SWBAT write a polynomial function given the real & complex zeros.

Find a 3rd degree polynomial that has the given zeros of: .{ 1, 3, 3}

* Complex numbers ALWAYS come in pairs. i.e. a+bi and a-bi.

Now let’s try together.Use whiteboards.

SWBAT write a polynomial function given the real & complex zeros.

Find a 5th degree polynomial that has the given zeros of: .{ 4,0,2,1 }i

* Complex numbers ALWAYS come in pairs. i.e. a+bi and a-bi.

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 4th degree polynomial that has the given zeros of: .{ 1,0, , }i i

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 2nd degree polynomial that has the given zeros of: .{ 2, 2}

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 4th degree polynomial that has the given zeros of: .{ 1,2, 3 }i

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 4th degree polynomial that has the given zeros of: .{5, 5,4 }i

Now you try.Write in notebook.

SWBAT write a polynomial function given the real & complex zeros.

Find a 2nd degree polynomial that has the given zeros of: .{9 , 9 }i i

Assignment:Write a polynomial function with real coefficients that has

the given zeros:

1.{9 , 9 } degree of 2

2. 0,2,3 0 has multiplicity of 3, degree of 5

3. 4,3 ,3 degree of 3

4. 3, 3 degree of 2

5. 2, 4 2 has multiplicity of 2, degree of 4

i i

i i

i