1
Unit 2
The previous functionbuilding showed how the factors of a polynomial determine its key characteristics. From the factors, you can determine the type and location of a polynomial’s zeros. Algebraic reasoning often allows you to reverse processes and work backwards.
Division of Polynomials
3
The Fundamental Theorem of Algebra states that every polynomial equations of degree must have roots. This means that every polynomial can be written as the product of factors.
For example:
You know that a factor of an integer divides into that integer with a remainder of zero.
Example:
Using division can help determine other factors. The factors of 115 are . In the same manner, factors of polynomials also divide into a polynomial without a remainder.
Review of Division
Or
Review of Polynomial Terms for Division
Dividend
Divisor
Quotient
5
Division of Polynomials using the
Area Model
Method
Example 1: Divide
Steps:
1. Create a box.
2. Write the denominator down the left side of the box
7 Example 3: Divide
What are the quotient and remainder?
Create a Area model (box)
Example 5:
Create a Area model (box)If you are ready to roll, start on EP 2.22
9
# 3
# 13
We will do problem 8 after you have spent
time working the rest of EP 2.22
Remainders
If you are lucky enough to get a remainder of
zero when dividing, then the divisor
divides
evenly
into the dividend.
This means that the divisor is a
factor
of the
dividend.
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Part B. Understanding Division in parts:
Determine the unknown.
Example 9
:Identify the divisor and the
remainder of
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Example 11: A rectangular pool has an area of
. The length of one side is . What is the length of the other side?
Check Core Standard Mastery:
Example 12: Julio claims that the
Using Division to solve a Problem
Example 13: The volume of the rectangular prism shown is given by the function cubic inches. If the length indicated is inches, find an equation for the shaded area in square inches. Remember Volume is length times width times height or .
Part C:The Factors of Life: The Factor
Theorem and Remainder Theorem
Remember from your experiences with division
that:
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It follows that any polynomial, , can be
written in the form:
Or
Generally, the linear factor is represent by the form
Given as the linear factor,
evaluate . Given and
Part C example 1: Divide the following Polynomial
If a polynomial is by
a linear factor , then the
is or the value of the equation when
Remainder Theorem
Example 14: Given and
23 Factor Theorem
A
polynomial
has a linear
polynomial
as a
factor
if and only if the remainder is
zero;
has as a
factor if and only
if .
Example 15. Haley and Lillian each prove that is a factor of the polynomial .
Haley Lillian
Example 16a: Is a factor of ?
Example 16b: Find what is for the function
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Example 21: Given the information: , and
a) b)
Sum of Cubes
Difference of Cubes
Reason Sum and Difference of Cubes works is Division…. Show how to always get the
pattern……….
Example 23: Factor the following polynomial,
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Example 22: Factor the following polynomial,
when
is a factor.
Example 24: Factor the following polynomial,
Example 25: Factor the following polynomial,
when
is a factor.
Sum of Cubes
31 Difference of Cubes
