Creating Polynomial Equations
The
factors and zeros
determine the
general shape of a
cubic and quartic
function
. The
table summarizes all
possible combinations
of roots and
What do we know????
What can we figure out?????
Given the graph, create an equation for the polynomial.
Example 5:
Equation:
Is it an even or odd polynomial?
How many zeros?
Do zeros match end behavior?
Leading coefficient positive or negative?
Do we have multiplicity? Do zeros touch or cross?
Complex Conjugate Root Theorem
If is a of a function with real , the
conjugate is also a of the .
Always comes in Pairs
Find the Conjugate for the Following Examples
Example 7: Example 8: Example 9:
A polynomial equation has the given roots. Find two Additional roots. What is the degree?
Example 10: Example 11: Example 12:
A third degree polynomial has the given roots.
Example 13: Example 14: Example 15:
Multiply the Complex Conjugates (What undoes
multiplication?)
Remember all
Complex
Roots
come with
Creating Polynomial Equations given Zeros/roots information
Given the roots, find the factors and write the polynomial equation in standard form.
Given the roots, find the factors and write the polynomial equation in standard form.
Example 2: Zero's:
Given the roots, find the factors and write the polynomial equation in standard form.
Given the roots, find the factors and write the polynomial equation in standard form.
Example 4: Roots:
What do we know????
Creating Polynomial Equations with part of the information given. Using understanding of Complex Roots to the other information. Example 20: The complex zeros of the cubic function are . Create the equation for the graph. Creating Polynomial Equations with part of the information given. Using understanding of Complex Roots to the other information.
Example 22: A cubic function has the roots .
Name the other root?
Create an equation for the function.
Example 23: A fourth degree polynomial has the roots of
. Name the other roots?
Example 23: A fourth degree polynomial has the roots of . Name the other roots?
Create an equation for the function.
Example: The length of each side of the cube is x units.
1. Write a function to describe the volume of the cube.
Select of few values for the length and find the volume.
Volumes of Different Shapes
Volumes of Different Shapes Cylinder ~ is a prism that has a circle as its base instead of a polyogon. Area of a Circle: Circumference of a Circle:
Volume Cylinder:
The PlantASeed Planter Company produces planterboxes To make the boxes, a square is cut from each
corner of a rectangular copper sheet. The sides are
bent to form a rectangular prism without a top. Cutting
different sized squares from the corners results in
different sized planter boxes. PlantASeed takes sales
orders from customers who request a sized planter box.
Each rectangular copper sheet is 12 inches by 18
inches. In the diagram, the solid lines indicate where
Complete the table. Include an expression for each planter box’s height, width, length, and volume for a square corner side of length h.
Building Volumes Equations with Polynomials
Example 24: A rectangular box is 2x+3 units long, 2x3 units wide, and 3x units high. What is its volume,
expressed in factored form?
Example 25: The length of a rectangle box is 3 inches greater that its width. The height of the box is 2 inches less than the width. Write a function that can model The volume of the situation.
Example 27: To make a rectangular prism, you
The PlantASeed Company also makes cylindrical
shaped planters for city sidewalks and store fronts The cylindrical shaped planters come in a variety of sizes, but all have a height to radius ratio of 2:1. Write a function
to represent the volume of the cylindrical planter in terms of the radius .
Example 28: For a cylinder, it has the following
Binomial Expansion
Rewrite each problem in expanded form
Expanded form: Expanded form: Expanded form:
Describe how you find the coefficients of the binomial expansion for the ‘next’ expansion?
How does the number of terms in the expansion relate to the degree?
