SOLUTION: 70a^2b+27ab-9b Factor the trinomial completely

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Question 191477: 70a^2b+27ab-9b
Factor the trinomial completely
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




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Looking at we can see that the first term is and the last term is where the coefficients are 70 and -9 respectively.

Now multiply the first coefficient 70 and the last coefficient -9 to get -630. Now what two numbers multiply to -630 and add to the middle coefficient 27? Let's list all of the factors of -630:



Factors of -630:
1,2,3,5,6,7,9,10,14,15,18,21,30,35,42,45,63,70,90,105,126,210,315,630

-1,-2,-3,-5,-6,-7,-9,-10,-14,-15,-18,-21,-30,-35,-42,-45,-63,-70,-90,-105,-126,-210,-315,-630 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -630
(1)*(-630)
(2)*(-315)
(3)*(-210)
(5)*(-126)
(6)*(-105)
(7)*(-90)
(9)*(-70)
(10)*(-63)
(14)*(-45)
(15)*(-42)
(18)*(-35)
(21)*(-30)
(-1)*(630)
(-2)*(315)
(-3)*(210)
(-5)*(126)
(-6)*(105)
(-7)*(90)
(-9)*(70)
(-10)*(63)
(-14)*(45)
(-15)*(42)
(-18)*(35)
(-21)*(30)

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


Now which of these pairs add to 27? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 27

First NumberSecond NumberSum
1-6301+(-630)=-629
2-3152+(-315)=-313
3-2103+(-210)=-207
5-1265+(-126)=-121
6-1056+(-105)=-99
7-907+(-90)=-83
9-709+(-70)=-61
10-6310+(-63)=-53
14-4514+(-45)=-31
15-4215+(-42)=-27
18-3518+(-35)=-17
21-3021+(-30)=-9
-1630-1+630=629
-2315-2+315=313
-3210-3+210=207
-5126-5+126=121
-6105-6+105=99
-790-7+90=83
-970-9+70=61
-1063-10+63=53
-1445-14+45=31
-1542-15+42=27
-1835-18+35=17
-2130-21+30=9



From this list we can see that -15 and 42 add up to 27 and multiply to -630


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



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So our expression goes from and factors further to


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Answer:

So completely factors to

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