document.write( "Question 131292: When 35z^2-23z-72 is factored completely, one of the factors is:\r
\n" ); document.write( "\n" ); document.write( "a. (7z-9)
\n" ); document.write( "b. (5z+8)
\n" ); document.write( "c. (7z+9)
\n" ); document.write( "d. (5z-9) \n" ); document.write( "

Algebra.Com's Answer #95888 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"35z%5E2-23z-72\"$ we can see that the first term is $\"35z%5E2\"$ and the last term is $\"-72\"$ where the coefficients are 35 and -72 respectively.\r
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\n" ); document.write( "\n" ); document.write( "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:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -2520:\r
\n" ); document.write( "\n" ); document.write( "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\r
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\n" ); document.write( "\n" ); document.write( "-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\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -2520\r
\n" ); document.write( "\n" ); document.write( "(1)*(-2520)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-1260)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-840)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-630)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-504)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-420)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-360)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-315)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-280)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-252)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-210)\r
\n" ); document.write( "\n" ); document.write( "(14)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(15)*(-168)\r
\n" ); document.write( "\n" ); document.write( "(18)*(-140)\r
\n" ); document.write( "\n" ); document.write( "(20)*(-126)\r
\n" ); document.write( "\n" ); document.write( "(21)*(-120)\r
\n" ); document.write( "\n" ); document.write( "(24)*(-105)\r
\n" ); document.write( "\n" ); document.write( "(28)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(30)*(-84)\r
\n" ); document.write( "\n" ); document.write( "(35)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(36)*(-70)\r
\n" ); document.write( "\n" ); document.write( "(40)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(42)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(45)*(-56)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(2520)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(1260)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(840)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(630)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(504)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(420)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(360)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(315)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(280)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(252)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(210)\r
\n" ); document.write( "\n" ); document.write( "(-14)*(180)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(168)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(140)\r
\n" ); document.write( "\n" ); document.write( "(-20)*(126)\r
\n" ); document.write( "\n" ); document.write( "(-21)*(120)\r
\n" ); document.write( "\n" ); document.write( "(-24)*(105)\r
\n" ); document.write( "\n" ); document.write( "(-28)*(90)\r
\n" ); document.write( "\n" ); document.write( "(-30)*(84)\r
\n" ); document.write( "\n" ); document.write( "(-35)*(72)\r
\n" ); document.write( "\n" ); document.write( "(-36)*(70)\r
\n" ); document.write( "\n" ); document.write( "(-40)*(63)\r
\n" ); document.write( "\n" ); document.write( "(-42)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-45)*(56)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "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\r
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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

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 40 and -63 add up to -23 and multiply to -2520\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"35z%5E2-23z-72\"$ , replace $\"-23z\"$ with $\"40z%2B-63z\"$ (notice $\"40z%2B-63z\"$ adds up to $\"-23z\"$ . So it is equivalent to $\"-23z\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"35z%5E2%2Bhighlight%2840z%2B-63z%29%2B-72\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"35z%5E2%2B40z-63z-72\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2835z%5E2%2B40z%29%2B%28-63z-72%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5z%287z%2B8%29-9%287z%2B8%29\"$ Factor out the GCF of $\"5z\"$ out of the first group. Factor out the GCF of $\"-9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285z-9%29%287z%2B8%29\"$ Since we have a common term of $\"7z%2B8\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"35z%5E2%2B40z-63z-72\"$ factors to $\"%285z-9%29%287z%2B8%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"35z%5E2-23z-72\"$ factors to $\"%285z-9%29%287z%2B8%29\"$ (since $\"35z%5E2-23z-72\"$ is equivalent to $\"35z%5E2%2B40z-63z-72\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"35z%5E2-23z-72\"$ factors to $\"%285z-9%29%287z%2B8%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So one of the factors of $\"35z%5E2-23z-72\"$ is D) $\"5z-9\"$ \n" ); document.write( "

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