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Mathematics LibreTexts

7.1E: Exercises

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    30550
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    Practice Makes Perfect

    Find the Greatest Common Factor of Two or More Expressions

    In the following exercises, find the greatest common factor.

    Exercise 1

    \(8,\; 18\)

    Answer

    \(2\)

    Exercise 2

    \(24,\; 40\)

    Exercise 3

    \(72,\; 162\)

    Answer

    \(18\)

    Exercise 4

    \(150,\; 275\)

    Exercise 5

    \(10a, \;50\)

    Answer

    \(10\)

    Exercise 6

    \(5b, \;30\)

    Exercise 7

    \(3x\), \(10x^2\)

    Answer

    \(x\)

    Exercise 8

    \(21b^2\), \(14b\)

    Exercise 9

    \(8w^2\), \(24w^3\)

    Answer

    \(8w^2\)

    Exercise 10

    \(30x^2\), \(18x^3\)

    Exercise 11

    \(10p^{3}q\), \(12pq^2\)

    Answer

    \(2pq\)

    Exercise 12

    \(8a^{2}b^3\), \(10ab^2\)

    Exercise 13

    \(12m^{2}n^3\), \(30m^{5}n^3\)

    Answer

    \(6m^{2}n^3\)

    Exercise 14

    \(28x^{2}y^4\), \(42x^{4}y^4\)

    Exercise 15

    \(10a^3\), \(12a^2\), 14a

    Answer

    \(2a\)

    Exercise 16

    \(20y^3\), \(28y^2\), 40y

    Exercise 17

    \(35x^3\), \(10x^4\), \(5x^5\)

    Answer

    \(5x^3\)

    Exercise 18

    \(27p^2\), \(45p^3\), \(9p^4\)

    Factor the Greatest Common Factor from a Polynomial

    In the following exercises, factor the greatest common factor from each polynomial.

    Exercise 19

    \(4x+20\)

    Answer

    4(x+5)

    Exercise 20

    \(8y+16\)

    Exercise 21

    \(6m+9\)

    Answer

    \(3(2m+3)\)

    Exercise 22

    \(14p+35\)

    Exercise 23

    \(9q+9\)

    Answer

    \(9(q+1)\)

    Exercise 24

    \(7r+7\)

    Exercise 25

    \(8m−8\)

    Answer

    \(8(m−1)\)

    Exercise 26

    \(4n−4\)

    Exercise 27

    \(9n−63\)

    Answer

    \(9(n−7)\)

    Exercise 28

    \(45b−18\)

    Exercise 29

    \(3x^2+6x−9\)

    Answer

    \(3(x^2+2x−3)\)

    Exercise 30

    \(4y^2+8y−4\)

    Exercise 31

    \(8p^2+4p+2\)

    Answer

    \(2(4p^2+2p+1)\)

    Exercise 32

    \(10q^2+14q+20\)

    Exercise 33

    \(8y^3+16y^2\)

    Answer

    \(8y^{2}(y+2)\)

    Exercise 34

    \(12x^3−10x\)

    Exercise 35

    \(5x^3−15x^2+20x\)

    Answer

    \(5x(x^2−3x+4)\)

    Exercise 36

    \(8m^2−40m+16\)

    Exercise 37

    \(12xy^2+18x^{2}y^2−30y^3\)

    Answer

    \(6y^{2}(2x+3x^2−5y)\)

    Exercise 38

    \(21pq^2+35p^{2}q^2−28q^3\)

    Exercise 39

    \(−2x−4\)

    Answer

    \(−2(x+2)\)

    Exercise 40

    \(−3b+12\)

    Exercise 41

    \(5x(x+1)+3(x+1)\)

    Answer

    \((x+1)(5x+3)\)

    Exercise 42

    \(2x(x−1)+9(x−1)\)

    Exercise 43

    \(3b(b−2)−13(b−2)\)

    Answer

    \((b−2)(3b−13)\)

    Exercise 44

    \(6m(m−5)−7(m−5)\)

    ​​​​​​Factor by Grouping

    In the following exercises, factor by grouping.

    Exercise 45

    \(xy+2y+3x+6\)

    Answer

    \((y+3)(x+2)\)

    Exercise 46

    \(mn+4n+6m+24\)

    Exercise 47

    \(uv−9u+2v−18\)

    Answer

    \((u+2)(v−9)\)

    Exercise 48

    \(pq−10p+8q−80\)

    Exercise 49

    \(b^2+5b−4b−20\)

    Answer

    \((b−4)(b+5)\)

    Exercise 50

    \(m^2+6m−12m−72\)

    Exercise 51

    \(p^2+4p−9p−36\)

    Answer

    \((p−9)(p+4)\)

    Exercise 52

    \(x^2+5x−3x−15\)

    Mixed Practice

    In the following exercises, factor.

    Exercise 53

    \(−20x−10\)

    Answer

    \(−10(2x+1)\)

    Exercise 54

    \(5x^3−x^2+x\)

    Exercise 55

    \(3x^3−7x^2+6x−14\)

    Answer

    \((x^2+2)(3x−7)\)

    Exercise 56

    \(x^3+x^2−x−1\)

    Exercise 57

    \(x^2+xy+5x+5y\)

    Answer

    \((x+y)(x+5)\)

    Exercise 58

    \(5x^3−3x^2−5x−3\)

    Everyday Math

    Exercise 59

    Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression \(w^2−6w\), where \(w=\) width. Factor the greatest common factor from the polynomial.

    Answer

    \(w(w−6)\)

    Exercise 60

    Height of a baseball The height of a baseball \(t\) seconds after it is hit is given by the expression \(−16t^2+80t+4\)

    Writing Exercises

    Exercise 61

    The greatest common factor of 36 and 60 is 12. Explain what this means.

    Answer

    Answers will vary.

    Exercise 62

    What is the GCF of \(y^4\), \(y^5\), and \(y^{10}\)? Write a general rule that tells you how to find the GCF of \(y^a\), \(y^b\), and \(y^c\).

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

    b. If most of your checks were:

    …confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

    …with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.