Question 131292: When 35z^2-23z-72 is factored completely, one of the factors is:
a. (7z-9)
b. (5z+8)
c. (7z+9)
d. (5z-9)
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 35 and -72 respectively.
Now multiply the first coefficient 35 and the last coefficient -72 to get -2520. Now what two numbers multiply to -2520 and add to the middle coefficient -23? Let's list all of the factors of -2520:
Factors of -2520:
1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520
-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-12,-14,-15,-18,-20,-21,-24,-28,-30,-35,-36,-40,-42,-45,-56,-60,-63,-70,-72,-84,-90,-105,-120,-126,-140,-168,-180,-210,-252,-280,-315,-360,-420,-504,-630,-840,-1260,-2520 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -2520
(1)*(-2520)
(2)*(-1260)
(3)*(-840)
(4)*(-630)
(5)*(-504)
(6)*(-420)
(7)*(-360)
(8)*(-315)
(9)*(-280)
(10)*(-252)
(12)*(-210)
(14)*(-180)
(15)*(-168)
(18)*(-140)
(20)*(-126)
(21)*(-120)
(24)*(-105)
(28)*(-90)
(30)*(-84)
(35)*(-72)
(36)*(-70)
(40)*(-63)
(42)*(-60)
(45)*(-56)
(-1)*(2520)
(-2)*(1260)
(-3)*(840)
(-4)*(630)
(-5)*(504)
(-6)*(420)
(-7)*(360)
(-8)*(315)
(-9)*(280)
(-10)*(252)
(-12)*(210)
(-14)*(180)
(-15)*(168)
(-18)*(140)
(-20)*(126)
(-21)*(120)
(-24)*(105)
(-28)*(90)
(-30)*(84)
(-35)*(72)
(-36)*(70)
(-40)*(63)
(-42)*(60)
(-45)*(56)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to -23? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -23
| First Number | Second Number | Sum | | 1 | -2520 | 1+(-2520)=-2519 | | 2 | -1260 | 2+(-1260)=-1258 | | 3 | -840 | 3+(-840)=-837 | | 4 | -630 | 4+(-630)=-626 | | 5 | -504 | 5+(-504)=-499 | | 6 | -420 | 6+(-420)=-414 | | 7 | -360 | 7+(-360)=-353 | | 8 | -315 | 8+(-315)=-307 | | 9 | -280 | 9+(-280)=-271 | | 10 | -252 | 10+(-252)=-242 | | 12 | -210 | 12+(-210)=-198 | | 14 | -180 | 14+(-180)=-166 | | 15 | -168 | 15+(-168)=-153 | | 18 | -140 | 18+(-140)=-122 | | 20 | -126 | 20+(-126)=-106 | | 21 | -120 | 21+(-120)=-99 | | 24 | -105 | 24+(-105)=-81 | | 28 | -90 | 28+(-90)=-62 | | 30 | -84 | 30+(-84)=-54 | | 35 | -72 | 35+(-72)=-37 | | 36 | -70 | 36+(-70)=-34 | | 40 | -63 | 40+(-63)=-23 | | 42 | -60 | 42+(-60)=-18 | | 45 | -56 | 45+(-56)=-11 | | -1 | 2520 | -1+2520=2519 | | -2 | 1260 | -2+1260=1258 | | -3 | 840 | -3+840=837 | | -4 | 630 | -4+630=626 | | -5 | 504 | -5+504=499 | | -6 | 420 | -6+420=414 | | -7 | 360 | -7+360=353 | | -8 | 315 | -8+315=307 | | -9 | 280 | -9+280=271 | | -10 | 252 | -10+252=242 | | -12 | 210 | -12+210=198 | | -14 | 180 | -14+180=166 | | -15 | 168 | -15+168=153 | | -18 | 140 | -18+140=122 | | -20 | 126 | -20+126=106 | | -21 | 120 | -21+120=99 | | -24 | 105 | -24+105=81 | | -28 | 90 | -28+90=62 | | -30 | 84 | -30+84=54 | | -35 | 72 | -35+72=37 | | -36 | 70 | -36+70=34 | | -40 | 63 | -40+63=23 | | -42 | 60 | -42+60=18 | | -45 | 56 | -45+56=11 |
From this list we can see that 40 and -63 add up to -23 and multiply to -2520
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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Answer:
So factors to
So one of the factors of is D) 
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