math 3 unit 4 day 2...

Math 3 Unit 4 Day 2 Notes.notebook October 20, 2015

Polynomial Functions

Unit 4 Day 2 from intercepts


Desmos Activity RecapThe factors of a polynomial function tells us its x-intercepts.

x-intecepts happen when y = 0 (For which x does f(x)=0?)

standard form

intercept form

x-intercepts

roots of the equation zeros of the function{


Some factors in a function have multiplicity.The multiplicity of a factor is how many times that factor appears in a polynomial.

standard form

intercept form

roots/zeros

x-intercepts

{multiplicity 2 multiplicity 3


multiplicity graph behavior examples

even touch

odd cross


The Fundamental Theorem of AlgebraA polynomial equation of degree n has exactly n roots in the set of complex numbers, including repeated roots.

• We know how many roots a polynomial has.

• If a polynomial is factorable, we can find the roots by factoring.

• If we graph the function, we can see the real roots they correspond to the xintercepts.

Implications:

(We'll work with factoring again next time. Yay!)

(Imaginary roots are not revealed by graphing... more about that later.)

(The degree tells us how many!)


If I give you the roots (or show you a picture of the graph), can you find an equaon for the polynomial?

By the end of class, for sure!

Today's big thing:


Example 1: Write a cubic polynomial function that has zeros 1, 2, and 3.


Look at the graph in Desmos...

https://www.desmos.com/calculator/rdqdizvl37

https://www.desmos.com/calculator/rdqdizvl37


Example 2: Write an equation for the cubic polynomial shown. (both intercept and standard forms...)

Intercept Form:

Standard Form:

Factors:Multiply:

check y-intercept:

x-intercepts:


Example 3: The three roots of cubic polynomial function f(x) are 0 and 3 with multiplicity 2. The function has a leading coefficient of 1/3. What is f(x)?


Example 4: Write an equation for the cubic function shown.


Practice A: Write a polynomial function that has zeros 1, 3, and 5.


Practice B: Write an equation (in standard form)for the cubic polynomial shown.


Practice C: Cubic function g(x) has two unique roots: 2 (mult. 1) and 1 (mult. 2). Function g(x) has a yintercept of 1/2. What is the standard form of g(x)?


Practice D: Write an equation for a quartic polynomial with zeros at 2, 0, 3, and 4.

• What is the y-intercept of your function?

math 3 unit 4 day 2...

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