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Name:
UNIT 3 LEARNING GUIDE – POLYNOMIALS
Instructions:
Using a pencil, complete the following questions as you work through the related lessons. Show
ALL work as it is explained in the lessons. Do your best and ask your instructor if you do not
understand any questions!
3.1 FACTORING REVIEW
1. Factor each polynomial.
a) 4?!? + 20?? − 16?!
) 5?!? − 10?? − 20?!
c) 6?ℎ! + 15?!ℎ" − 9?ℎ#
d) −40?!?# − 8?!?! − 16?"?#
e) ?(3? + 1) − 2(3? + 1)
f) ?(? − 4) + 8(? − 4)
2. . Factor each trinomial
a) ?! + 10? + 16
) ?! + 12? + 35
c) ?! + ? − 2
d) ?! − 6? + 8
e) ?! + 4? − 12
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3. Factor the following polynomials
a) 4?! − 1
) 36?! − ?!
c) 25?! − 144ℎ!
d) 625 − ?!
e) 2500 − 81?!
f) ?# −?!
4. Utilize all of the strategies for factoring in order to factor the following polynomials
a) ?! − 4? − 2? + 8
) −10? + ?! − XXXXXXXXXX?
c) (2?! + 23?) − (3? − 18)
d) 5?? + 3??! − 2?!? − 7??! + 3??
e) 4?! + 10 + ? − 50 − ?! XXXXXXXXXX?
f) −100 + 9?! + 19 − 5?!
g) −7? + 5?! + 6 − 3?! − 11? + 30
h) 2?? + 30?! − ?? − 16?# − ?? − 5?!
5. Utilize all of the strategies for factoring in order to factor the following polynomials.
a) ?! + 2? + 1 − ?!
) 4?! − ?! − 10? − 25
c) ?" + 2?! + 6? + 12
d) ?! − 5? + 3?? − 15?
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XXXXXXXXXXREVIEW MESSY TRINOMIALS
1. 1. Factor each of the following completely. Show your work using the procedural method. Verify the
factoring is co
ect by mentally multiplying the factors
a) 2x2 + x – 1 b) 2x2 + 11x + 12
c) 4x2 + 9x XXXXXXXXXXd) 6x2 – 13x + 6
e) 6x2 – 17x XXXXXXXXXXf) 12x2 + x – 6
g) 4x2 – 16x XXXXXXXXXXh) 12x2 – 5x – 25
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2. Factor each of the following completely. Show your work using the procedural method if the trinomial is
messy.
a) 4x2 + 14x XXXXXXXXXXb) 8x2 + 28x – 16
c) 12x2 + 24xy + 9y2 d) $
!
x2 + %
!
x + 6
e) 3x2 – 3x + !
"
f) 0.1x2 + 0.7x + 1
3. Factor each of the following completely. It may be necessary to rea
ange the terms first.
a) –14 – 15x – 4x2 b) 34x + 24x2 + 7
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c) 2x4 + 7x XXXXXXXXXXd) 8x4 – 2x2 – 3
e) 3x4y2 + 10x2y XXXXXXXXXXf) 10x4 + 37x2y + 7y2
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XXXXXXXXXXSOLVING BY FACTORING
1. Solve each of the following by factoring.
a) m2 + 2m = 0 b) m2 + 9m + 18 = 0
c) b2 + 12b + 32 = 0 d) 7p2 – 19p – 6 = 0
e) 5n2 – 6n + 1 = 0 f) 7p2 + 39p – 18 = 0
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2. Solve each of the following by factoring. You will need to rea
ange the terms first to create a
trinomial.
a) 7b2 + 70b + 176 = 8 b) k2 – 10k = –16
XXXXXXXXXXTHE REMAINDER THEOREM
1. In your own words explain the Remainder Theorem.
2. Using either long or synthetic division, determine the division statement for the
following questions.
a) (x3 + 4x2 − 6x − 9) ÷ (x − 2) b) (7x3 − 9x2 + 3x) ÷ (x − 1)
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3. Use the remainder theorem to determine the remainder when each of the following
polynomials is divided by the binomial (x + 4).
a) P(x) = x2 − 3x + 5 b) P(x) = 3x2 + 10x − 8
c) P(x) = x3 − 7x + 6 d) P(x) = −x3 + 5x2 − x + 12
e) Which of the above polynomials has a root of –4? Explain
4. Use the remainder theorem to determine the remainder for each of the following divisions.
a) (x2 − 7x) ÷ (x + 6) b) (x2 + 4x + 9) ÷ (x − m)
c) (x3 − x + 4) ÷ (x + 2) d) (2m3 + 3m2 − 5m − 10) ÷ (2m − 1)
5. a) The volume of an open top box is given by V(x) = x3 + 2x2 − 5x − 6 and the length can
e expressed as (x + 1). Determine the factors that describe the width and height. Assume
the width is less than the height.
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) What are the dimensions of the box if x equals 7 ft?
6. Given the remainder, determine the value of m for each of the following operations.
a) P(x) = (x2 + mx − 7) ÷ (x − 1); remainder = 5
) P(x) = (x3 + mx2 + 6x – 3) ÷ (x + 3); remainder = –21
c) P(x) = (x4 − 3x3 + x2 + mx + 5) ÷ (2x − 1); remainder = 47
16
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7. When the polynomial P(x) = 2x3 + mx2 + nx − 5 is divided by (x − 1) the remainder is 1. When
the polynomial is divided by (x + 2) the remainder is –5. Determine the values of both m and
n.
8. P(x) = 7x3 + mx2 + nx + 2. When P(x) is divided by (x + 1), the remainder is 6. When P(x)
is divided by (x – 2), the remainder is 42. Determine the values of both m and n.
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3.5 THE FACTOR THEOREM
1. Explain in your own words how to determine all the potential rational roots for a
polynomial and then do so for the polynomials below.
a) P(x) = x3 + 2x2 − 7x − 4
) P(x) = 2x3 − 5x2 + 3
c) P(x) = −3x3 + 2x2 − 7x + 6
2. Could (x − 3) be a factor of the polynomial P(x) = 2x3 + 6x + 5? Why or why not? Show
your work.
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3. Use the factor theorem to determine whether each of the following polynomials has