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Question 139589This question is from textbook Algebra Structure and Method
: I have been struggling with this problem for awhile now and I was wondering if anyone could help me? I would deeply appreciate it! Please and Thank you!

Factoring Patterns for ax^2+bx+c
Factor. Check by multiplying the factors. If the polynomial is not factorable, write prime.
8+45r-18r^2 This question is from textbook Algebra Structure and Method

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
8%2B45r-18r%5E2 Start with the given expression


-18r%5E2%2B45r%2B8 Sort the terms in descending order


-%2818r%5E2-45r-8%29 Factor out a negative one


Looking at the inner polynomial 18r%5E2-45r-8, we can see that the first term is 18r%5E2 and the last term is -8

Now multiply the first coefficient 18 and the last term -8 to get -144. Now what two numbers multiply to -144 and add to the middle coefficient -45? Let's list all of the factors of -144:



Factors of -144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -144
(1)*(-144)
(2)*(-72)
(3)*(-48)
(4)*(-36)
(6)*(-24)
(8)*(-18)
(9)*(-16)
(-1)*(144)
(-2)*(72)
(-3)*(48)
(-4)*(36)
(-6)*(24)
(-8)*(18)
(-9)*(16)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -45? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -45

First NumberSecond NumberSum
1-1441+(-144)=-143
2-722+(-72)=-70
3-483+(-48)=-45
4-364+(-36)=-32
6-246+(-24)=-18
8-188+(-18)=-10
9-169+(-16)=-7
-1144-1+144=143
-272-2+72=70
-348-3+48=45
-436-4+36=32
-624-6+24=18
-818-8+18=10
-916-9+16=7



From this list we can see that 3 and -48 add up to -45 and multiply to -144


Now looking at the expression 18r%5E2-45r-8, replace -45r with 3r%2B-48r (notice 3r%2B-48r adds up to -45r. So it is equivalent to -45r)

18r%5E2%2Bhighlight%283r%2B-48r%29%2B-8


Now let's factor 18r%5E2%2B3r-48r-8 by grouping:


%2818r%5E2%2B3r%29%2B%28-48r-8%29 Group like terms


3r%286r%2B1%29-8%286r%2B1%29 Factor out the GCF of 3r out of the first group. Factor out the GCF of -8 out of the second group


%283r-8%29%286r%2B1%29 Since we have a common term of 6r%2B1, we can combine like terms

So 18r%5E2%2B3r-48r-8 factors to %283r-8%29%286r%2B1%29


So this also means that 18r%5E2-45r-8 factors to %283r-8%29%286r%2B1%29 (since 18r%5E2-45r-8 is equivalent to 18r%5E2%2B3r-48r-8)



So -%2818r%5E2-45r-8%29 factors to -%283r-8%29%286r%2B1%29

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Answer:

So 8%2B45r-18r%5E2 factors to -%283r-8%29%286r%2B1%29