8-5
Dividing Polynomials
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SOLVE IT!
S
In how many ways is it possible to replace the squares with single-digit numbers to
complete a correct division problem? Justify your answer.
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Exploration
Vocabulary
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Use Representations to Communicate Mathematical Ideas (1)(E)
Describe how the representation you used to solve the problem successfully
organizes and communicates your ideas.
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Problem 1
P
Got It?
Using Polynomial Long Division
Use polynomial long division to divide 3x3 - 8x2 - 21x + 56 by x2 - 7. What are
the quotient and remainder?
Learning
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TEKS Process Standard (1)(G)
Problem 2
P
Got It?
Using Polynomial Long Division to Check Factors
a. Is x 2 - 1 a factor of P (x) = x 4 + 4x 2 - 5? If it is, write P (x) as the product of
two factors.
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#
b. Use the fact that 12 31 = 372 to write the polynomial 3x 2 + 7x + 2 as the
product of two factors.
Problem 3
P
Got It?
Using Synthetic Division
Use synthetic division to divide x4 - 49x2 - 8x + 56 by x - 7. What are the
quotient and remainder?
Lesson 8-5 | Dividing Polynomials
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Animation
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TEKS Process Standard (1)(A)
Problem 4
P
Got It?
Using Synthetic Division to Solve a Problem
If the polynomial x 3 + 6x 2 + 11x + 6 expresses the volume, in cubic inches, of a
box, and the width is x + 1 in., what are the dimensions of the box?
ELPS
ELPS
EL
Learning
Animation
Read the Got It with a classmate. Use a rectangular box as visual support for
solving the problem. What do the expressions in the problem represent for the
box? What dimensions are missing? How do you solve for the missing dimensions?
Problem 5
P
Got It?
Evaluating a Polynomial
Given that P (x) = x 4 - 3x 3 - 28x 2 + 5x + 20, what is P( -4)?
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Lesson Check
Do you know HOW?
1. Divide x3 + 5x2 + 11x + 15 by x + 3 using any method.
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2. When a polynomial is divided by 5x + 7, the quotient is 2x3 - 4x + 2 with
remainder 1. Find the polynomial.
3. The remainder from the division of the polynomial 3x3 + 6x2 - rx + 18 by
x + 2 is 26. What is the value of r?
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Lesson Check
Do you UNDERSTAND?
4. Analyze Mathematical Relationships (1)(F) A polynomial P(x) is divided
by a binomial x - a. The remainder is 0. What conclusion can you draw?
Explain your answer.
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5. Use a Problem-Solving Model (1)(B) Explain why it is important to have
the terms of both polynomials written in descending order of degree
before dividing.
6. Create Representations to Communicate Mathematical Ideas (1)(E) Write
a polynomial division that has a quotient of x + 3 and a remainder of 2.
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TEXAS Test Practice
T
Gridded Response
Solve each exercise and enter your answer in the grid provided.
1. The volume of a box is given by 4x3 + 8x2 + x - 3, and its height is given by
x + 1. Calculate the area of the bottom of the box (that is, the length times
width). What is the leading coefficient of the polynomial representing the
area of the box?
2. What is the missing value in the following synthetic division?
-4
0
1
-5
-4
1
11
-4
4
12
-44
160
-40
172
3. What is the remainder when you divide x4 - 9x3 + 27x2 - 27x by x - 3?
4. How many unique factors does x4 + 4x3 - 3x2 - 14x - 8 have,
including x + 4?
1.
2.
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0
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0
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