Download Section 5: Polynomials – PART 1

Section 5: Polynomials – PART 1
The following Mathematics Florida Standards will be covered
in this section:
MAFS.912.A-APR.1.1 Understand that polynomials form a
system analogous to the integers;
namely, they are closed under the
operations of addition, subtraction,
and multiplication; add, subtract, and
multiply polynomials.
MAFS.912.A-APR.3.4 Prove polynomial identities and use
them to describe numerical
relationships.
MAFS.912.A-SSE.1.2 Use the structure of an expression to
identify ways to rewrite it. For example,
see x^4– y^4 as (x^2)^2 – (y^2)^2, thus
recognizing it as a difference of
squares that can be factored as (x^2 –
y^2)(x^2 + y^2).
Videos in this Section
Video 1:
Video 2:
Video 3:
Video 4:
Video 5:
Video 6:
Introduction to Polynomials – Part 1
Introduction to Polynomials – Part 2
Adding and Subtracting Polynomials
Multiplying Polynomials
Polynomials Identities – Part 1
Polynomials Identities – Part 2
Section 5: Polynomials – PART 1
1
Section 5 – Video 1
Introduction to Polynomials – Part 1
A polynomial is a finite sum of terms in which all variables are
raised to nonnegative integer powers and no variables
appear in any denominator.
Determine whether each of the following expressions is a
polynomial. If the expressions are not polynomials, justify your
reasoning.
+
+
+
+
+
−
+
+
Section 5: Polynomials – PART 1
2
Classifying Polynomials
We can classify polynomials by the number of terms.
Number
of Terms
Example
7
+
+
+
+
+
Name of Polynomial
+
We can also classify polynomials by degree.
Classifying Polynomials
Degree
Example
Type of Polynomial
+
+
+
Section 5: Polynomials – PART 1
3
Let’s Practice!
Classify each polynomial below by degree and by number of
terms.
−
−
Try It!
Classify each polynomial below by degree and by number of
terms.
−
+
+
+
+
Section 5: Polynomials – PART 1
4
We can also apply the closure property to polynomials.
A set is said to be _________ under a specific mathematical
operation if the ________ that occurs when you perform the
operation on any two members of the set is also a member of
the set.
Determine whether each of the following statements is true or
false. If a statements is false, write a counterexample.
Integers are closed under addition.
Odd numbers are closed under addition.
Even numbers are closed under addition.
Negative numbers are closed under multiplication.
Odd numbers are closed under multiplication.
Section 5: Polynomials – PART 1
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When referring to the closure property, what do you think
“polynomials form a system analogous to the integers”
means?
Determine whether each of the following statements is true or
false. If the statement is false, write a counterexample.
Polynomials are closed under addition.
Polynomials are closed under subtraction.
Polynomials are closed under multiplication.
Polynomials are closed under division.
Section 5: Polynomials – PART 1
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Section 5 – Video 2
Introduction to Polynomials – Part 2
At times, we may use function notation to represent
polynomials.
For example, we might use � � to represent the polynomial
expression � + � − and write the polynomial function,
� � = � + �− .
To find �
we would substitute
expression and evaluate. Find �
for � in the polynomial
.
Let’s Practice!
If
� =� +�+
and
� = � − , find the following.
( )
Section 5: Polynomials – PART 1
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Try It!
Write a polynomial function, � � , of degree
�
= .
such that
We often see applications of polynomial functions in the real
world.
Let’s Practice!
The function � = . � + . � + . can be used to
approximate the snack and beverage sales by a national
vendor, where � is the number of years since 1980 and � is
sales, in millions of dollars.
Approximate the vendor snack and beverage sales in 2012.
Use the function to predict the vendor snack and beverage
sales in 2018.
Section 5: Polynomials – PART 1
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Try It!
Rainbow Bridge is a natural arch at the base of the Navajo
Mountains. With a height of
feet, it is often described as
the world’s largest natural bridge. Assuming no air resistance,
the height of an object dropped from the bridge is given by
the polynomial function �
=−
+
at seconds.
Find the height of the object at =
and =
Section 5: Polynomials – PART 1
seconds.
9
BEAT THE TEST
1. Two functions are given below.
� =� + � − �+
� =�
Candice solved
�
�
as follows below.
� + � − �+
�
�
�
�
+
−
+
�
�
�
�
�+ −
�
+
�
Part A: Candice’s work illustrates that polynomials are
o closed
o not closed
under
o
o
o
o
addition
division
multiplication
subtraction
.
Part B: Justify your answer from Part A.
Section 5: Polynomials – PART 1
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2.
A moving company changed the size of its most popular
moving box. The volume of the new box, � � , can be
represented by � � = � + � + � − , where � is the size
of each dimension of the old box, in inches.
Part A: Which of the following explains what �
represents in this context?
A �
B
C
D
represents the volume of the new box provided
the old box had dimensions ” x ” x ”.
�
represents the volume of the old box provided
the new box has dimensions ” x ” x ”.
�
represents the dimensions of the old box
provided that the new box has a volume of
cubic
inches.
�
represents the dimensions of the new box
provided that the old box had a volume of
cubic
inches.
Part B: If the old box has a side
inches long, what is the
difference between the volume of the new box
and the volume of the old box?
cubic inches.
Section 5: Polynomials – PART 1
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Section 5 – Video 3
Adding and Subtracting Polynomials
Consider
=
+
− and
=
−
Write an expression to represent
+
.
Write an expression to represent
−
..
Write an expression to represent [
Section 5: Polynomials – PART 1
]+ [
+ .
].
12
Let’s Practice!
Two polynomial functions are given.
= (
−
=
Find
+
+
7
+
−
−
− )
.
Section 5: Polynomials – PART 1
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Try It!
Two polynomial functions are given.
= (
ℎ
Find
=(
−
+
+
−ℎ
−
− )
+
+
)
.
A practical application of subtracting polynomials is finding
the profit function �
for a business when given the revenue
function �
and a cost function � .
Let’s Practice!
� =
�
=�
−�
−�
The cost function for a company to produce armbands is
�
=
+
, where is the number of armbands. The
company sells the armbands for $ each. Write a revenue
function and the profit function for the company.
Section 5: Polynomials – PART 1
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Try It!
A company uses two different types of shipping boxes and
the owners need to determine how much more space the
larger box has for shipping purposes. The volume of the
smaller box can be represented by the function
=
+
+
+ . The volume of the larger box can
be represented by the function
=
+
+
+ .
Write an expression to represent the extra space required for
the larger box.
Section 5: Polynomials – PART 1
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BEAT THE TEST!
1. Consider the following polynomial functions:
�
�
ℎ �
�
�
�
=− � +
+ �
=− � − � −
= � − − �
= −� + � +
=− � + � −
= � + � −
Which of the following could result in equivalent
expressions?
A
B
C
D
I and II
II and III
I and III
I, II, and III
I. � +
II. ℎ � −
III. � −
�
�
�
Section 5: Polynomials – PART 1
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Section 5 – Video 4
Multiplying Polynomials
Let’s extend our understanding of the the distributive property
by learning how to multiply polynomials.
Let’s Practice!
Multiply the following polynomials and write an equivalent
expression.
− �
� −
+
( � − )( � − )
Section 5: Polynomials – PART 1
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� + �−
−
�+
+
Section 5: Polynomials – PART 1
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Try It!
Multiply the following polynomials and write an equivalent
expression.
−
−
+ �
− �
−
Section 5: Polynomials – PART 1
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BEAT THE TEST!
1. Consider the figure below.
– � ��
� –
��
�–
� – � – � +
��
��
What is the total area, in square feet, of the above figure?
Section 5: Polynomials – PART 1
20
Section 5 – Video 5
Polynomial Identities – Part 1
Let’s look at visual representations of various polynomial
identities.
Let’s Practice!
Consider the figure below.
Find the area of the figure by dividing it into regions.
Section 5: Polynomials – PART 1
21
Try It!
Consider the figure below.
b
a
b
a
Find the area for the figure by dividing it into regions.
Section 5: Polynomials – PART 1
22
Let’s Practice!
Let’s think about how to find the volume of this cube with the
corner taken out. Write an expression to represent the volume
of the cube.
Write a numeric expression to represent the total volume of
the figures below.
Write equivalent expressions for the volume of the two
images.
Section 5: Polynomials – PART 1
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Try It!
Write an algebraic expression to represent the volume for the
figure below.
b
a
b
b
b
a
a
a
b
b
a
a
Next, let’s split the cubes apart. Write algebraic expressions to
represent the total volume of the figures below.
a
b
b
b
a
a
Write equivalent expressions for the volume of the two
images.
Section 5: Polynomials – PART 1
24
Section 5 – Video 6
Polynomial Identities – Part 2
A polynomial identity is a true equation, that is generalized, so
that it can apply to multiple situations.
Based on this definition, we can infer that both expressions
are always ________________, no matter what number
replaces the variables.
Show that
+
=
+
+
Use the polynomial identity +
=
+
equivalent expressions for the following.
+
to write
+
+
Section 5: Polynomials – PART 1
25
Common Polynomial Identities:
 Difference of two squares:
−
±
 Perfect square trinomials:
 Sum and difference of cubes:
+
 Quadratic formula: If
Let’s Practice!
Prove
−
+
=
=
+
±
−
=
=
±
+ = , then
+
for any
±
+
=
∓
− ±√ 2 −
+
and .
Pythagorean triples are _____________ solutions to the
Pythagorean Theorem.
How can we use this polynomial identity to generate
Pythagorean triples?
Use the polynomial identity to generate a set of Pythagorean
triples.
Section 5: Polynomials – PART 1
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Try It!
+
Prove
−
+
=
+
for any
and .
Let’s Practice!
Use polynomial identities to factor and write equivalent
expressions for each of the following polynomials.
−
−
Section 5: Polynomials – PART 1
27
Try It!
Use polynomial identities to factor and write equivalent
expressions for each of the following polynomials.
−
+
+
Section 5: Polynomials – PART 1
28
BEAT THE TEST!
1.
Consider the figure below.
Part A: Write a numeric expression to represent the
volume of the figure.
Part B: What polynomial identity is represented by the
figure?
Section 5: Polynomials – PART 1
29
2. Use polynomial identities to match the expressions on the
left with their equivalent expression on the right.
1. ________
2. ________
3. ________
4. ________
5. ________
+
+
−
−
−
+
+
A
B
C
D
E
Section 5: Polynomials – PART 1
−
+
+
+
−
+
+
−
+
−
30