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Answers
Teacher Copy
Lesson 15-3
Dividing Polynomials
Plan
Pacing: 1 class period
Chunking the Lesson
Example A #1 Example B
p. 248
Example C #2
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Students should recall that an absolute value of a number is its distance from zero on a number line.
Have students evaluate the following:
1. |6| [6]
2. |–6| [6]
Then have students solve the following equation.
3. |x|= 6 [x = 6 or x = –6]
Example A Marking the Text, Interactive Word Wall
©
2014
College
Board.
All rights
reserved.
Point
out the
Math Tip
to reinforce
why two
solutions exist. Work through the solutions to the equation algebraically. Remind students that
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solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check
their results.
Developing Math Language
An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving
algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they
contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two
numbers that have a specific distance from zero on a number line.
1 Identify a Subtask, Quickwrite
When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute
value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary.
Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to
write each equation and then discuss how the solution set is represented by the graph.
Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation
Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler
example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could
cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by
implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line.
Have groups present their findings to the class.
ELL Support
For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different
ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a
negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think
of it.
Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount
(greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).
Developing Math Language
An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <,
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>, ≤, ≥, or ≠. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the
solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.
Example C Simplify the Problem, Debriefing
Before addressing Example C, discuss the following: Inequalities with |A| > b, where b is a positive number, are known as disjunctions
and are written as A < –b or A > b.
For example, |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x
< – 5 or x > 5.
See graph A.
This also holds true for |A| ≥ b.
Inequalities with |A| < b, where b is a positive number, are known as conjunctions and are written as –b < A < b, or as –b < A and A < b.
For example: |x| < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution
is –5 < x < 5.
See graph B.
This also holds true for |A| ≤ b.
Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic
equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the
whole class.
Teacher to Teacher
Another method for solving inequalities relies on the geometric definition of absolute value |x – a| as the distance from x to a. Here’s how
you can solve the inequality in the example:
Thus, the solution set is all values of x whose distance from is greater than . The solution can be represented on a number line and written
as x < –4 or x > 1.
2 Quickwrite, Self Revision/Peer Revision, Debriefing
Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving
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inequalities.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of
students present their solutions to Item 4.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson
concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the
activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute
value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having
difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities.
Activity Standards Focus
In Activity 15, students learn how to perform operations with polynomials including addition, subtraction, multiplication, long division,
and synthetic division. Because polynomials may have several terms, emphasize to students the importance of performing polynomial
operations carefully so that no terms are skipped
Plan
Pacing: 1 class period
Chunking the Lesson
#1–5 #6–7
Check Your Understanding
#12–13
Check Your Understanding
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#19–21
Lesson Practice
Teach
Bell-Ringer Activity
Ask students to evaluate the function f(x) = 2x3 − 3x2 + 1 for each value.
1. f(0) [f(0) = 1]
2. f(1) [f(1) = 0]
3. f(−3) [f(−3) = −80]
4. f(7) [f(7) = 540]
5. f(1.5) [f(1.5) = 1]
Teacher to Teacher
This guided exploration introduces students to polynomial operations, initially set in a real-world context. Review the meanings of
terms such as revenue and operating costs as needed to help students understand the scenario. The context has a discrete domain in
order to keep calculations simple and to allow explorations through tables, graphs, and analytic methods. However, students should
be comfortable with adding continuous functions as well.
1–5 Discussion Groups, Create Representations, Think-Pair-Share, Self Revision/Peer Revision, Debriefing
Have groups brainstorm and find methods for solutions, such as by using a table, a graph, or analytic methods. Choose groups that
represent a spectrum of solution methods to share with the entire class. Monitor presentations to ensure that students are using
appropriate words and clearly explaining their solution methods. For Item 5, ask students to justify that their answer is reasonable.
Common Core State Standards for Activity 15
HSA-APR.A.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.
HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.
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HSA-APR.D.6: Rewrite simple rational expressions in diff erent forms; write
in the form q(x) +
, where a(x), b(x),
q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the
more complicated examples, a computer algebra system.
6–7 Create Representations, Debriefing
Item 7 is designed to help students see the connection between a graphic representation of the sum of two polynomial functions
and the process of adding two functions algebraically. The round dots represent the values of the function K(t). The square dots
represent the values of the function F(t). Students should be able to use the graph to plot the points for S(t). If students do not see
how to find the points of S(t) graphically, help them make the connection by adding the value of one function to the value of the
other at every point.
Differentiating Instruction
Some students may need a table of values for all 12 months to correctly graph S(t). If so, have them create and complete a new
table for months 6-12.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand within the context of the problem why S(t) is greater than
both K(t) and F(t) at every point. Ask students whether this will always be true when adding two polynomial functions.
12–13 Create Representations, Discussion Groups, Debriefing
The same context is used to introduce students to polynomial subtraction. Ask students to suggest a title for the table used in Item
13, and be sure students can describe what each column in the table represents. Students should be able to easily translate between
algebraic representations and context.
Have student groups discuss and answer the following questions:
During which month was revenue the greatest? Give a possible explanation why that month’s revenue is the greatest.
What might account for the fluctuation in costs each month?
Would you say that Polly’s business is successful based on the information in the table? Explain.
Encourage all group members to share their ideas and opinions. Groups should listen to all members before reaching consensus.
ELL Support
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To support students in reading problem scenarios, carefully group students to ensure that all students participate and have an
opportunity for meaningful reading and discussion. Suggest that group members each read an item and explain what that item
means to them. Group members can then confirm one another’s understanding of key information.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand the table. Ask students how the information in the table
could be used to make predictions for future months.
19–21 Create Representations, Discussion Groups, Debriefing
Have groups answer these questions and discuss why the second quarter might produce the greatest profits. Then have students
present a plan for increasing profits in the other quarters, explaining why they think their plan will work. Ensure that students
use appropriate terminology in their presentations, and remind them to use transitions to help communicate how one thought
moves into another.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the
lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination
for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand the meaning of profit and loss as well as revenue.
If not, provide a review of these and other basic business terms including operating costs, fixed costs, and variable costs.
Plan
Pacing: 1-2 class periods
Chunking the Lesson
Example A
Check Your Understanding
#5 #6
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Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Combine like terms.
1. 7x + 15x − 2 + 8 + 12 [22x + 18]
2. −5x + 6x − 2x + 4 − 11 [−x − 7]
3. 5 + 14x − 8x + 14 + x [7x + 19]
4. x + 5x + 6 − 10 + 8 − 6x − 4 [0]
Example A Marking the Text, Note Taking, Debriefing
The Try These items are useful as a formative assessment. For those students having difficulty with addition and subtraction
of polynomials, some intervention may be necessary. Use vertical addition and subtraction to focus attention on grouping and
combining like terms. Have students mark the text to help identify and group like terms. Use additional practice and
one-on-one assistance, as needed.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand concepts related to adding and subtracting
polynomials. In Item 2, be sure students distribute the negative sign over the entire expression in parentheses.
Developing Math Language
The definition of standard form is revisited in this lesson. Review the definition from Activity 14 and use the Bell-Ringer
Activity to show students that combining like terms is an essential step when writing a polynomial in standard form.
5 Think-Pair-Share
This item relates the sums and differences of polynomials back to the context. This connection can also be analyzed using the
table and graphing features of a graphing calculator to help further the concept of adding and subtracting polynomials with
graphs and tables.
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6 Think-Pair-Share, Debriefing
This item extends the concept of the Distributive Property to polynomials with more than two terms. Mini-Lesson: Use a
Graphic Organizer to Multiply Polynomials can help students keep track of partial products.
Universal Access
Sometimes it is difficult for students to see what was done from one step to the next when simplifying an expression. Ask
students to identify what has changed from one line to the next. This may help them understand what step was taken to get
there.
MINI-LESSON: Use a Graphic Organizer to Multiply Polynomials
If students need additional help multiplying polynomials, a mini-lesson is available to provide practice.
See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.
Technology Tip
Students can use tables and graphs on a graphing calculator to check their answers when performing polynomial
operations. For example, in Item 7, students can enter (x + 5)(x2 + 4x − 5) as y1 and their answer as y2. If the answer is
correct, a table will show the same values for all values of x, and the graphs will coincide.
For additional technology resources, visit SpringBoard Digital.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand the different methods for multiplying polynomials.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the
lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a
culmination for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand how to perform addition, subtraction, and
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multiplication of polynomials. Have students describe in words the process used for each. If students make errors when
performing operations, encourage them to use visual cues (e.g., using different colors to circle like terms, drawing
arrows to show distribution) as they work through each problem.
Plan
Pacing: 1 class period
Chunking the Lesson
#1 Example A #2–3
Check Your Understanding
Example B
Check Your Understanding
Lesson Practice
Teach
Bell-Ringer Activity
Simplify.
1. [a]
2. [xy2]
3. [x + 1]
4. a3b2]
1 Activating Prior Knowledge
The process of long division of integers is similar to that of long division of polynomials. Working through this problem
will help students see the similarities in the division algorithms and make the process more comfortable.
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Example A Note Taking
Remind students that in long division of polynomials, when subtracting the product of the first term of the quotient
and the divisor, the difference should be zero. If it is not, then students need to look at the factors they are multiplying
to identify the error.
Differentiating Instruction
Support students who have difficulty with Example A by relating the process to division involving integers. Work the
problem
side by side with the problem
so students can see the parallels between the two methods.
2–3 Create Representations, Discussion Groups, Debriefing
Students have the opportunity to work through these problems using Example A as support. Students will benefit from
guided practice with long division before working on the Check Your Understanding problems independently.
Monitor discussions to ensure that mathematical concepts such as distributing, grouping, and combining like terms are
being verbalized precisely.
Teacher to Teacher
It may be helpful for students to look at and explain the end behavior of the quotient they found in Item 2. This will be
a good prelude to investigating rational functions and end behavior in later units. It relates to the concept of limits and
asymptotes while not developing those concepts explicitly. This will give students an advantage when they begin to
look at a more complex analysis of these functions.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand polynomial division. Watch for students who
do not recognize that Items 5 and 7 are division problems that require long division to solve.
Example B Marking the Text, Debriefing
Students can mark the text with additional arrows to help them understand the algorithm for synthetic division.
Differentiating Instruction
Have students look at the value of f(3) in the solution to Example B. Have them make a conjecture about the
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relationship between the remainder when they divide a polynomial by (x − 3) and the value of f(3).
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand the algorithm for synthetic division and that
they can explain why it works. Relate synthetic division to long division.
Check Your Understanding
Debrief students’ answers to these items to ensure that they understand Pythagorean triples and how to generate
them using the algorithm.
Assess
Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding
of the lesson concepts and their ability to apply their learning.
See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a
culmination for the activity.
Adapt
Check students’ answers to the Lesson Practice to ensure that they understand that the division problems can be
solved using either long division or synthetic division. Common student errors include omitting terms with
coefficients of 0 (such as 0x2 in Item 14a) and subtracting instead of adding while performing synthetic division. If
students make these kinds of errors, have them write down the steps for long division and synthetic division and
refer to these steps each time they solve a problem.
Learning Targets
p. 248
Determine the quotient of two polynomials.
Prove a polynomial identity and use it to describe numerical relationships.
Note Taking (Learning Strategy)
Definition
Creating a record of information while reading a text or listening to a speaker
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Purpose
Helps in organizing ideas and processing information
Marking the Text (Learning Strategy)
Definition
Highlighting, underlining, and /or annotating text to focus on key information to help understand the text
or solve the problem
Purpose
Helps the reader identify important information in the text and make notes about the interpretation of tasks
required and concepts to apply to reach a solution
Create Representations (Learning Strategy)
Definition
Creating pictures, tables, graphs, lists, equations, models, and /or verbal expressions to interpret text or
data
Purpose
Helps organize information using multiple ways to present data and to answer a question or show a
problem solution
Discussion Groups (Learning Strategy)
Definition
Working within groups to discuss content, to create problem solutions, and to explain and justify a solution
Purpose
Aids understanding through the sharing of ideas, interpretation of concepts, and analysis of problem
scenarios
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Suggested Learning Strategies
Note Taking, Marking the Text, Create Representations, Discussion Groups
Polynomial long division has a similar algorithm to numerical long division.
1. Use long division to find the quotient
.
Example A
p. 249
Divide x3 − 7x2 + 14 by x − 5, using long division.
Step 1:
Set up the division problem with the divisor and dividend written in descending order of degree. Include
zero coefficients for any missing terms.
Step 2:
Divide the first term of the dividend [x3] by the first term of the divisor [x].
Step 3:
Multiply the result [x2] by the divisor [x2(x − 5) = x3 − 5x2].
Step 4:
Subtract to get a new result [−2x2 + 0x + 14].
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Math Tip
When the division process is complete, the degree of the remainder will be less than the degree of
the divisor.
Step 5:
Repeat the steps.
Solution:
Try These A
Use long division to find each quotient.
a. (x3 − x2 − 6x + 18) ÷ (x + 3)
x2 − 4x + 6
b.
x2 + 3x + 3
When a polynomial function f(x) is divided by another polynomial function d(x), the outcome is a
new quotient function consisting of a polynomial p(x) plus a remainder function r(x).
Connect to Technology
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You can use a CAS (computer algebra system) to perform division on more complicated
quotients.
2. Follow the steps from Example A to find the quotient of
3. Find the quotient of
.
.
−4x
Check Your Understanding
p. 250
Use long division to find each quotient. Show your work.
4. (x2 + 5x − 3) ÷ (x − 5)
5.
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6. (x3 − 9) ÷ (x + 3)
7.
Synthetic division is another method of polynomial division that is useful when the divisor has
the form x − k.
Example B
p. 251
Divide x4 − 13x2 + 32 by x − 3 using synthetic division.
Step 1:
Set up the division problem using only coefficients for the dividend and only the constant for the
divisor. Include zero coefficients for any missing terms [x3 and x].
Step 2:
Bring down the leading coefficient [1].
Step 3:
Multiply the coefficient [1] by the divisor [3]. Write the product [1 • 3 = 3] under the second
coefficient [0] and add [0 + 3 = 3].
Step 4:
Repeat this process until there are no more coefficients.
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Step 5:
The numbers in the bottom row become the coefficients of the quotient. The number in the last
column is the remainder. Write it over the divisor.
Solution:
Try These B
Math Tip
Remember, when using synthetic division, the divisor must be in the form x − k. When the
divisor is in the form x + k, write it as x − (−k) before you begin the process.
Use long division to find each quotient.
a.
x2 − x − 6
b.
Check Your Understanding
8. Use synthetic division to divide
.
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9. In synthetic division, how does the degree of the quotient compare to the degree of
the dividend?
The degree of the quotient is always one less than the degree of the dividend.
10. Construct viable arguments. Justify the following statement: The set of
polynomials is closed under addition, subtraction, and multiplication, but not under
division.
The set of polynomials is closed under addition, subtraction, and multiplication because the
result of performing these operations is a polynomial. However, division of polynomials
often leads to a quotient with a remainder. The remainder becomes a rational function,
which means the set of polynomials is not closed under division.
There are a number of polynomial identities that can be used to describe important
p. 252
numerical relationships in math. For example, the polynomial identity (x2 + y2)2 = (x2 −
y2)2 + (2xy)2 can be used to generate a famous numerical relationship that is used in
geometry.
Math Tip
When verifying an identity, choose one side of the equation to work with and try to
make that side look like the other side.
First, let’s verify the identity using what we have learned in this lesson about polynomial
operations.
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Now that we have verified the identity, let’s see how it relates to a famous numerical
relationship. If we evaluate it for x = 2 and y = 1, we get:
The numbers 3, 4, and 5 are known as Pythagorean triples because they fit the condition
a2 + b2 = c2, which describes the lengths of the legs and hypotenuse of a right triangle.
Thus, the polynomial identity (x2 + y2)2 = (x2 − y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
Check Your Understanding
11. Use the polynomial identity above to generate a Pythagorean triple given x = 5 and
y = 2.
20,21, and 29
12. Use the polynomial identity to see what happens when the values of x and y are
the same. Does the identity generate a Pythagorean triple in this case? Use an
example to support your answer.
No; if the values of x and y are the same, the term (x2 − y2)2 = 0, and the identity does
not generate a Pythagorean triple.
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13. Reason abstractly. Are there any other specific values for x and y that would
not generate Pythagorean triples? If so, what value(s)?
Yes; 0
Lesson 15-3 Practice
14. Find each quotient using long division.
a. (6x3 + x − 1) ÷ (x + 2)
b.
2
−2x + x + 1
15. Make sense of problems. Find each quotient using synthetic division.
a. (x3 − 8) ÷ (x − 2)
2
x + 2x + 4
b.
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