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Mathematics LibreTexts

7.3E: Exercises

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    Practice Makes Perfect

    Recognize a Preliminary Strategy to Factor Polynomials Completely

    In the following exercises, identify the best method to use to factor each polynomial.

    Exercise 1

    1. \(10q^2+50\)
    2. \(a^2−5a−14\)
    3. \(uv+2u+3v+6\)
    Answer
    1. factor the GCF, binomial
    2. Undo FOIL
    3. factor by grouping

    Exercise 2

    1. \(n^2+10n+24\)
    2. \(8u^2+16\)
    3. \(pq+5p+2q+10\)

    Exercise 3

    1. \(x^2+4x−21\)
    2. \(ab+10b+4a+40\)
    3. \(6c^2+24\)
    Answer
    1. undo FOIL
    2. factor by grouping
    3. factor the GCF, binomial

    Exercise 4

    1. \(20x^2+100\)
    2. \(uv+6u+4v+24\)
    3. \(y^2−8y+15\)
    Factor Trinomials of the form \(ax^2+bx+c\) with a GCF

    In the following exercises, factor completely.

    Exercise 5

    \(5x^2+35x+30\)

    Answer

    \(5(x+1)(x+6)\)

    Exercise 6

    \(12s^2+24s+12\)

    Exercise 7

    \(2z^2−2z−24\)

    Answer

    \(2(z−4)(z+3)\)

    Exercise 8

    \(3u^2−12u−36\)

    Exercise 9

    \(7v^2−63v+56\)

    Answer

    \(7(v−1)(v−8)\)

    Exercise 10

    \(5w^2−30w+45\)

    Exercise 11

    \(p^3−8p^2−20p\)

    Answer

    \(p(p−10)(p+2)\)

    Exercise 12

    \(q^3−5q^2−24q\)

    Exercise 13

    \(3m^3−21m^2+30m\)

    Answer

    \(3m(m−5)(m−2)\)

    Exercise 14

    \(11n^3−55n^2+44n\)

    Exercise 15

    \(5x^4+10x^3−75x^2\)

    Answer

    \(5x^{2}(x−3)(x+5)\)

    Exercise 16

    \(6y^4+12y^3−48y^2\)

    Factor Trinomials Using Trial and Error

    In the following exercises, factor.

    Exercise 17

    \(2t^2+7t+5\)

    Answer

    \((2t+5)(t+1)\)

    Exercise 18

    \(5y^2+16y+11\)

    Exercise 19

    \(11x^2+34x+3\)

    Answer

    \((11x+1)(x+3)\)

    Exercise 20

    \(7b^2+50b+7\)

    Exercise 21

    \(4w^2−5w+1\)

    Answer

    \((4w−1)(w−1)\)

    Exercise 22

    \(5x^2−17x+6\)

    Exercise 23

    \(6p^2−19p+10\)

    Answer

    \((3p−2)(2p−5)\)

    Exercise 24

    \(21m^2−29m+10\)

    Exercise 25

    \(4q^2−7q−2\)

    Answer

    \((4q+1)(q−2)\)

    Exercise 26

    \(10y^2−53y−11\)

    Exercise 27

    \(4p^2+17p−15\)

    Answer

    \((4p−3)(p+5)\)

    Exercise 28

    \(6u^2+5u−14\)

    Exercise 29

    \(16x^2−32x+16\)

    Answer

    \(16(x−1)(x−1)\)

    Exercise 30

    \(81a^2+153a−18\)

    Exercise 31

    \(30q^3+140q^2+80q\)

    Answer

    \(10q(3q+2)(q+4)\)

    Exercise 32

    \(5y^3+30y^2−35y\)

    Factor Trinomials using the ‘ac’ Method

    In the following exercises, factor.

    Exercise 33

    \(5n^2+21n+4\)

    Answer

    \((5n+1)(n+4)\)

    Exercise 34

    \(8w^2+25w+3\)

    Exercise 35

    \(9z^2+15z+4\)

    Answer

    \((3z+1)(3z+4)\)

    Exercise 36

    \(3m^2+26m+48\)

    Exercise 37

    \(4k^2−16k+15\)

    Answer

    \((2k−3)(2k−5)\)

    Exercise 38

    \(4q^2−9q+5\)

    Exercise 39

    \(5s^2−9s+4\)

    Answer

    \((5s−4)(s−1)\)

    Exercise 40

    \(4r^2−20r+25\)

    Exercise 41

    \(6y^2+y−15\)

    Answer

    \((3y+5)(2y−3)\)

    Exercise 42

    \(6p^2+p−22\)

    Exercise 43

    \(2n^2−27n−45\)

    Answer

    \((2n+3)(n−15)\)

    Exercise 44

    \(12z^2−41z−11\)

    Exercise 45

    \(3x^2+5x+4\)

    Answer

    prime

    Exercise 46

    \(4y^2+15y+6\)

    Exercise 47

    \(60y^2+290y−50\)

    Answer

    \(10(6y−1)(y+5)\)

    Exercise 48

    \(6u^2−46u−16\)

    Exercise 49

    \(48z^3−102z^2−45z\)

    Answer

    \(3z(8z+3)(2z−5)\)

    Exercise 50

    \(90n^3+42n^2−216n\)

    Exercise 51

    \(16s^2+40s+24\)

    Answer

    \(8(2s+3)(s+1)\)

    Exercise 52

    \(24p^2+160p+96\)

    Exercise 53

    \(48y^2+12y−36\)

    Answer

    \(12(4y−3)(y+1)\)

    Exercise 54

    \(30x^2+105x−60\)

    ​​​​​Mixed Practice

    In the following exercises, factor.

    Exercise 55

    \(12y^2−29y+14\)

    Answer

    \((4y−7)(3y−2)\)

    Exercise 56

    \(12x^2+36y−24z\)

    Exercise 57

    \(a^2−a−20\)

    Answer

    \((a−5)(a+4)\)

    Exercise 58

    \(m^2−m−12\)

    Exercise 59

    \(6n^2+5n−4\)

    Answer

    \((2n−1)(3n+4)​​\)

    Exercise 60

    \(12y^2−37y+21\)

    Exercise 61

    \(2p^2+4p+3\)

    Answer

    prime

    Exercise 62

    \(3q^2+6q+2\)

    Exercise 63

    \(13z^2+39z−26\)

    Answer

    \(13(z^2+3z−2)\)

    Exercise 64

    \(5r^2+25r+30\)

    Exercise 65

    \(x^2+3x−28\)

    Answer

    \((x+7)(x−4)\)

    Exercise 66

    \(6u^2+7u−5\)

    Exercise 67

    \(3p^2+21p\)

    Answer

    \(3p(p+7)\)

    Exercise 68

    \(7x^2−21x\)

    Exercise 69

    \(6r^2+30r+36\)

    Answer

    \(6(r+2)(r+3)\)

    Exercise 70

    \(18m^2+15m+3\)

    Exercise 71

    \(24n^2+20n+4\)

    Answer

    \(4(2n+1)(3n+1)\)

    Exercise 72

    \(4a^2+5a+2\)

    Exercise 73

    \(x^2+2x−24\)

    Answer

    \((x+6)(x−4)\)

    Exercise 74

    \(2b^2−7b+4\)

    Everyday Math

    Exercise 75

    Height of a toy rocket The height of a toy rocket launched with an initial speed of \(80\) feet per second from the balcony of an apartment building is related to the number of seconds, \(t\), since it is launched by the trinomial \(−16t^2+80t+96\). Factor this trinomial.

    Answer

    \(−16(t−6)(t+1)\)

    Exercise 76

    Height of a beach ball The height of a beach ball tossed up with an initial speed of \(12\) feet per second from a height of \(4\) feet is related to the number of seconds, \(t\), since it is tossed by the trinomial \(−16t^2+12t+4\). Factor this trinomial.

    Writing Exercises

    Exercise 77

    List, in order, all the steps you take when using the “\(ac\)” method to factor a trinomial of the form \(ax^2+bx+c\).

    Answer

    Answers may vary.

    Exercise 78

    How is the “\(ac\)” method similar to the “undo FOIL” method? How is it different?

    Exercise 79

    What are the questions, in order, that you ask yourself as you start to factor a polynomial? What do you need to do as a result of the answer to each question?

    Answer

    Answers may vary.

    Exercise 80

    On your paper draw the chart that summarizes the factoring strategy. Try to do it without looking at the book. When you are done, look back at the book to finish it or verify it.

    ​​​​​​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 row is “recognize a preliminary strategy to factor polynomials completely”. The second row is “factor trinomials of the form a x ^ 2 + b x + c with a GCF”. The third row is “factor trinomials using trial and error”. And the fourth row is “factor trinomials using the “ac” method”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

    b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?