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# 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. $$uv+2u+3v+6$$
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. $$6c^2+24$$
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

$$5x^2+35x+30$$

$$5(x+1)(x+6)$$

Exercise 6

$$12s^2+24s+12$$

Exercise 7

$$2z^2−2z−24$$

$$2(z−4)(z+3)$$

Exercise 8

$$3u^2−12u−36$$

Exercise 9

$$7v^2−63v+56$$

$$7(v−1)(v−8)$$

Exercise 10

$$5w^2−30w+45$$

Exercise 11

$$p^3−8p^2−20p$$

$$p(p−10)(p+2)$$

Exercise 12

$$q^3−5q^2−24q$$

Exercise 13

$$3m^3−21m^2+30m$$

$$3m(m−5)(m−2)$$

Exercise 14

$$11n^3−55n^2+44n$$

Exercise 15

$$5x^4+10x^3−75x^2$$

$$5x^{2}(x−3)(x+5)$$

Exercise 16

$$6y^4+12y^3−48y^2$$

Factor Trinomials Using Trial and Error

In the following exercises, factor.

Exercise 17

$$2t^2+7t+5$$

$$(2t+5)(t+1)$$

Exercise 18

$$5y^2+16y+11$$

Exercise 19

$$11x^2+34x+3$$

$$(11x+1)(x+3)$$

Exercise 20

$$7b^2+50b+7$$

Exercise 21

$$4w^2−5w+1$$

$$(4w−1)(w−1)$$

Exercise 22

$$5x^2−17x+6$$

Exercise 23

$$6p^2−19p+10$$

$$(3p−2)(2p−5)$$

Exercise 24

$$21m^2−29m+10$$

Exercise 25

$$4q^2−7q−2$$

$$(4q+1)(q−2)$$

Exercise 26

$$10y^2−53y−11$$

Exercise 27

$$4p^2+17p−15$$

$$(4p−3)(p+5)$$

Exercise 28

$$6u^2+5u−14$$

Exercise 29

$$16x^2−32x+16$$

$$16(x−1)(x−1)$$

Exercise 30

$$81a^2+153a−18$$

Exercise 31

$$30q^3+140q^2+80q$$

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

Exercise 32

$$5y^3+30y^2−35y$$

Factor Trinomials using the ‘ac’ Method

In the following exercises, factor.

Exercise 33

$$5n^2+21n+4$$

$$(5n+1)(n+4)$$

Exercise 34

$$8w^2+25w+3$$

Exercise 35

$$9z^2+15z+4$$

$$(3z+1)(3z+4)$$

Exercise 36

$$3m^2+26m+48$$

Exercise 37

$$4k^2−16k+15$$

$$(2k−3)(2k−5)$$

Exercise 38

$$4q^2−9q+5$$

Exercise 39

$$5s^2−9s+4$$

$$(5s−4)(s−1)$$

Exercise 40

$$4r^2−20r+25$$

Exercise 41

$$6y^2+y−15$$

$$(3y+5)(2y−3)$$

Exercise 42

$$6p^2+p−22$$

Exercise 43

$$2n^2−27n−45$$

$$(2n+3)(n−15)$$

Exercise 44

$$12z^2−41z−11$$

Exercise 45

$$3x^2+5x+4$$

prime

Exercise 46

$$4y^2+15y+6$$

Exercise 47

$$60y^2+290y−50$$

$$10(6y−1)(y+5)$$

Exercise 48

$$6u^2−46u−16$$

Exercise 49

$$48z^3−102z^2−45z$$

$$3z(8z+3)(2z−5)$$

Exercise 50

$$90n^3+42n^2−216n$$

Exercise 51

$$16s^2+40s+24$$

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

Exercise 52

$$24p^2+160p+96$$

Exercise 53

$$48y^2+12y−36$$

$$12(4y−3)(y+1)$$

Exercise 54

$$30x^2+105x−60$$

​​​​​Mixed Practice

In the following exercises, factor.

Exercise 55

$$12y^2−29y+14$$

$$(4y−7)(3y−2)$$

Exercise 56

$$12x^2+36y−24z$$

Exercise 57

$$a^2−a−20$$

$$(a−5)(a+4)$$

Exercise 58

$$m^2−m−12$$

Exercise 59

$$6n^2+5n−4$$

$$(2n−1)(3n+4)​​$$

Exercise 60

$$12y^2−37y+21$$

Exercise 61

$$2p^2+4p+3$$

prime

Exercise 62

$$3q^2+6q+2$$

Exercise 63

$$13z^2+39z−26$$

$$13(z^2+3z−2)$$

Exercise 64

$$5r^2+25r+30$$

Exercise 65

$$x^2+3x−28$$

$$(x+7)(x−4)$$

Exercise 66

$$6u^2+7u−5$$

Exercise 67

$$3p^2+21p$$

$$3p(p+7)$$

Exercise 68

$$7x^2−21x$$

Exercise 69

$$6r^2+30r+36$$

$$6(r+2)(r+3)$$

Exercise 70

$$18m^2+15m+3$$

Exercise 71

$$24n^2+20n+4$$

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

Exercise 72

$$4a^2+5a+2$$

Exercise 73

$$x^2+2x−24$$

$$(x+6)(x−4)$$

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.

$$−16(t−6)(t+1)$$

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 $$ax^2+bx+c$$.

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?