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7.2E: Exercises

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Practice Makes Perfect

Factor Trinomials of the Form $$x^2+bx+c$$

In the following exercises, factor each trinomial of the form $$x^2+bx+c$$

Exercise 1

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

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

Exercise 2

$$y^2+8y+7$$

Exercise 3

$$m^2+12m+11$$

$$(m+1)(m+11)$$

Exercise 4

$$b^2+14b+13$$

Exercise 5

$$a^2+9a+20$$

$$(a+4)(a+5)$$

Exercise 6

$$m^2+7m+12$$ ​​​​​

Exercise 7

$$p^2+11p+30$$

$$(p+5)(p+6)$$

Exercise 8

$$w^2+10w+21$$

Exercise 9

$$n^2+19n+48$$

$$(n+3)(n+16)$$

Exercise 10

$$b^2+14b+48$$

Exercise 11

$$a^2+25a+100$$

$$(a+5)(a+20)$$

Exercise 12

$$u^2+101u+100$$

Exercise 13

$$x^2−8x+12$$

$$(x−2)(x−6)$$

Exercise 14

$$q^2−13q+36$$

Exercise 15

$$y^2−18y+45$$

$$(y−3)(y−15)$$

Exercise 16

$$m^2−13m+30$$

Exercise 17

$$x^2−8x+7$$

$$(x−1)(x−7)$$

Exercise 18

$$y^2−5y+6$$

Exercise 19

$$p^2+5p−6$$

$$(p−1)(p+6)$$

Exercise 20

$$n^2+6n−7$$

Exercise 21

$$y^2−6y−7$$

$$(y+1)(y−7)$$

Exercise 22

$$v^2−2v−3$$

Exercise 23

$$x^2−x−12$$

$$(x−4)(x+3)$$

Exercise 24

$$r^2−2r−8$$

Exercise 25

$$a^2−3a−28$$

$$(a−7)(a+4)$$

Exercise 26

$$b^2−13b−30$$

Exercise 27

$$w^2−5w−36$$

$$(w−9)(w+4)$$

Exercise 28

$$t^2−3t−54$$

Exercise 29

$$x^2+x+5$$

prime

Exercise 30

$$x^2−3x−9$$

Exercise 31

$$8−6x+x^2$$

$$(x−4)(x−2)$$

Exercise 32

$$7x+x^2+6$$

Exercise 33

$$x^2−12−11x$$

$$(x−12)(x+1)$$

Exercise 34

$$−11−10x+x^2$$

​​​​​​Factor Trinomials of the Form $$x^2+bxy+cy^2$$

In the following exercises, factor each trinomial of the form $$x^2+bxy+cy^2$$

Exercise 33

$$p^2+3pq+2q^2$$

$$(p+q)(p+2q)$$

Exercise 34

$$m^2+6mn+5n^2$$

Exercise 35

$$r^2+15rs+36s^2$$

$$(r+3s)(r+12s)$$

Exercise 36

$$u^2+10uv+24v^2$$

Exercise 37

$$m^2−12mn+20n^2$$

$$(m−2n)(m−10n)$$

Exercise 38

$$p^2−16pq+63q^2$$

Exercise 39

$$x^2−2xy−80y^2$$

$$(x+8y)(x−10y)$$

Exercise 40

$$p^2−8pq−65q^2$$

Exercise 41

$$m^2−64mn−65n^2$$

$$(m+n)(m−65n)$$

Exercise 42

$$p^2−2pq−35q^2$$

Exercise 43

$$a^2+5ab−24b^2$$

$$(a+8b)(a−3b)$$

Exercise 44

$$r^2+3rs−28s^2$$

Exercise 45

$$x^2−3xy−14y^2$$

prime

Exercise 46

$$u^2−8uv−24v^2$$

Exercise 47

$$m^2−5mn+30n^2$$

prime

Exercise 48

$$c^2−7cd+18d^2$$

​​​​​​Mixed Practice

In the following exercises, factor each expression.

Exercise 49

$$u^2−12u+36$$

$$(u−6)(u−6)$$

Exercise 50

$$w^2+4w−32$$

Exercise 51

$$x^2−14x−32$$

$$(x+2)(x−16)$$

Exercise 52

$$y^2+41y+40$$

Exercise 53

$$r^2−20rs+64s^2$$

$$(r−4s)(r−16s)$$

Exercise 54

$$x^2−16xy+64y^2$$

Exercise 55

$$k^2+34k+120$$

$$(k+4)(k+30)$$

Exercise 56

$$m^2+29m+120$$

Exercise 57

$$y^2+10y+15$$

prime

Exercise 58

$$z^2−3z+28$$

Exercise 59

$$m^2+mn−56n^2$$

$$(m+8n)(m−7n)$$

Exercise 60

$$q^2−29qr−96r^2$$

Exercise 61

$$u^2−17uv+30v^2$$

$$(u−15v)(u−2v)$$

Exercise 62

$$m^2−31mn+30n^2$$

Exercise 63

$$c^2−8cd+26d^2$$

prime

Exercise 64

$$r^2+11rs+36s^2$$

Everyday Math

Exercise 65

Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial $$x^2+x−56$$ describes how these numbers are related. Factor the trinomial.

$$(x+8)(x−7)$$

Exercise 66

Consecutive integers Deshawn is thinking of two consecutive integers whose product is 182. The trinomial $$x^2+x−182$$ describes how these numbers are related. Factor the trinomial describes how these numbers are related. Factor the trinomial.

​​​​​​Writing Exercises

Exercise 67

Many trinomials of the form $$x^2+bx+c$$ factor into the product of two binomials $$(x+m)(x+n)$$. Explain how you find the values of $$m$$ and $$n$$.

Exercise 68

How do you determine whether to use plus or minus signs in the binomial factors of a trinomial of the form $$x^2+bx+c$$ where $$b$$ and $$c$$ may be positive or negative numbers?

Exercise 69

Will factored $$x^2−x−20$$ as $$(x+5)(x−4)$$. Bill factored it as $$(x+4)(x−5)$$. Phil factored it as $$(x−5)(x−4)$$. Who is correct? Explain why the other two are wrong.

Look at Example, where we factored $$y^2+17y+60$$. We made a table listing all pairs of factors of 60 and their sums. Do you find this kind of table helpful? Why or why not?