Question 148141: how do i factor 20c(squared+17c-63r squared
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 20 and -63 respectively.
Now multiply the first coefficient 20 and the last coefficient -63 to get -1260. Now what two numbers multiply to -1260 and add to the middle coefficient 17? Let's list all of the factors of -1260:
Factors of -1260:
1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,28,30,35,36,42,45,60,63,70,84,90,105,126,140,180,210,252,315,420,630,1260
-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-28,-30,-35,-36,-42,-45,-60,-63,-70,-84,-90,-105,-126,-140,-180,-210,-252,-315,-420,-630,-1260 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -1260
(1)*(-1260)
(2)*(-630)
(3)*(-420)
(4)*(-315)
(5)*(-252)
(6)*(-210)
(7)*(-180)
(9)*(-140)
(10)*(-126)
(12)*(-105)
(14)*(-90)
(15)*(-84)
(18)*(-70)
(20)*(-63)
(21)*(-60)
(28)*(-45)
(30)*(-42)
(35)*(-36)
(-1)*(1260)
(-2)*(630)
(-3)*(420)
(-4)*(315)
(-5)*(252)
(-6)*(210)
(-7)*(180)
(-9)*(140)
(-10)*(126)
(-12)*(105)
(-14)*(90)
(-15)*(84)
(-18)*(70)
(-20)*(63)
(-21)*(60)
(-28)*(45)
(-30)*(42)
(-35)*(36)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17
| First Number | Second Number | Sum | | 1 | -1260 | 1+(-1260)=-1259 | | 2 | -630 | 2+(-630)=-628 | | 3 | -420 | 3+(-420)=-417 | | 4 | -315 | 4+(-315)=-311 | | 5 | -252 | 5+(-252)=-247 | | 6 | -210 | 6+(-210)=-204 | | 7 | -180 | 7+(-180)=-173 | | 9 | -140 | 9+(-140)=-131 | | 10 | -126 | 10+(-126)=-116 | | 12 | -105 | 12+(-105)=-93 | | 14 | -90 | 14+(-90)=-76 | | 15 | -84 | 15+(-84)=-69 | | 18 | -70 | 18+(-70)=-52 | | 20 | -63 | 20+(-63)=-43 | | 21 | -60 | 21+(-60)=-39 | | 28 | -45 | 28+(-45)=-17 | | 30 | -42 | 30+(-42)=-12 | | 35 | -36 | 35+(-36)=-1 | | -1 | 1260 | -1+1260=1259 | | -2 | 630 | -2+630=628 | | -3 | 420 | -3+420=417 | | -4 | 315 | -4+315=311 | | -5 | 252 | -5+252=247 | | -6 | 210 | -6+210=204 | | -7 | 180 | -7+180=173 | | -9 | 140 | -9+140=131 | | -10 | 126 | -10+126=116 | | -12 | 105 | -12+105=93 | | -14 | 90 | -14+90=76 | | -15 | 84 | -15+84=69 | | -18 | 70 | -18+70=52 | | -20 | 63 | -20+63=43 | | -21 | 60 | -21+60=39 | | -28 | 45 | -28+45=17 | | -30 | 42 | -30+42=12 | | -35 | 36 | -35+36=1 |
From this list we can see that -28 and 45 add up to 17 and multiply to -1260
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
So this also means that factors to  (since is equivalent to )
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Answer:
So factors to
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