document.write( "Question 207384: There was not a topic for what I am looking for. My problem says factor the trinomial completely and I'm not sure how to do it.
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Algebra.Com's Answer #156815 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 the expression $\"54t%5E2%2B3t-70\"$ , we can see that the first coefficient is $\"54\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-70\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"54\"$ by the last term $\"-70\"$ to get $\"%2854%29%28-70%29=-3780\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-3780\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-3780\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-3780\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,27,28,30,35,36,42,45,54,60,63,70,84,90,105,108,126,135,140,180,189,210,252,270,315,378,420,540,630,756,945,1260,1890,3780\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-27,-28,-30,-35,-36,-42,-45,-54,-60,-63,-70,-84,-90,-105,-108,-126,-135,-140,-180,-189,-210,-252,-270,-315,-378,-420,-540,-630,-756,-945,-1260,-1890,-3780\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"-3780\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-3780) = -3780
\n" ); document.write( "2*(-1890) = -3780
\n" ); document.write( "3*(-1260) = -3780
\n" ); document.write( "4*(-945) = -3780
\n" ); document.write( "5*(-756) = -3780
\n" ); document.write( "6*(-630) = -3780
\n" ); document.write( "7*(-540) = -3780
\n" ); document.write( "9*(-420) = -3780
\n" ); document.write( "10*(-378) = -3780
\n" ); document.write( "12*(-315) = -3780
\n" ); document.write( "14*(-270) = -3780
\n" ); document.write( "15*(-252) = -3780
\n" ); document.write( "18*(-210) = -3780
\n" ); document.write( "20*(-189) = -3780
\n" ); document.write( "21*(-180) = -3780
\n" ); document.write( "27*(-140) = -3780
\n" ); document.write( "28*(-135) = -3780
\n" ); document.write( "30*(-126) = -3780
\n" ); document.write( "35*(-108) = -3780
\n" ); document.write( "36*(-105) = -3780
\n" ); document.write( "42*(-90) = -3780
\n" ); document.write( "45*(-84) = -3780
\n" ); document.write( "54*(-70) = -3780
\n" ); document.write( "60*(-63) = -3780
\n" ); document.write( "(-1)*(3780) = -3780
\n" ); document.write( "(-2)*(1890) = -3780
\n" ); document.write( "(-3)*(1260) = -3780
\n" ); document.write( "(-4)*(945) = -3780
\n" ); document.write( "(-5)*(756) = -3780
\n" ); document.write( "(-6)*(630) = -3780
\n" ); document.write( "(-7)*(540) = -3780
\n" ); document.write( "(-9)*(420) = -3780
\n" ); document.write( "(-10)*(378) = -3780
\n" ); document.write( "(-12)*(315) = -3780
\n" ); document.write( "(-14)*(270) = -3780
\n" ); document.write( "(-15)*(252) = -3780
\n" ); document.write( "(-18)*(210) = -3780
\n" ); document.write( "(-20)*(189) = -3780
\n" ); document.write( "(-21)*(180) = -3780
\n" ); document.write( "(-27)*(140) = -3780
\n" ); document.write( "(-28)*(135) = -3780
\n" ); document.write( "(-30)*(126) = -3780
\n" ); document.write( "(-35)*(108) = -3780
\n" ); document.write( "(-36)*(105) = -3780
\n" ); document.write( "(-42)*(90) = -3780
\n" ); document.write( "(-45)*(84) = -3780
\n" ); document.write( "(-54)*(70) = -3780
\n" ); document.write( "(-60)*(63) = -3780\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"3\"$ :\r
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First Number	Second Number	Sum
1	-3780	1+(-3780)=-3779
2	-1890	2+(-1890)=-1888
3	-1260	3+(-1260)=-1257
4	-945	4+(-945)=-941
5	-756	5+(-756)=-751
6	-630	6+(-630)=-624
7	-540	7+(-540)=-533
9	-420	9+(-420)=-411
10	-378	10+(-378)=-368
12	-315	12+(-315)=-303
14	-270	14+(-270)=-256
15	-252	15+(-252)=-237
18	-210	18+(-210)=-192
20	-189	20+(-189)=-169
21	-180	21+(-180)=-159
27	-140	27+(-140)=-113
28	-135	28+(-135)=-107
30	-126	30+(-126)=-96
35	-108	35+(-108)=-73
36	-105	36+(-105)=-69
42	-90	42+(-90)=-48
45	-84	45+(-84)=-39
54	-70	54+(-70)=-16
60	-63	60+(-63)=-3
-1	3780	-1+3780=3779
-2	1890	-2+1890=1888
-3	1260	-3+1260=1257
-4	945	-4+945=941
-5	756	-5+756=751
-6	630	-6+630=624
-7	540	-7+540=533
-9	420	-9+420=411
-10	378	-10+378=368
-12	315	-12+315=303
-14	270	-14+270=256
-15	252	-15+252=237
-18	210	-18+210=192
-20	189	-20+189=169
-21	180	-21+180=159
-27	140	-27+140=113
-28	135	-28+135=107
-30	126	-30+126=96
-35	108	-35+108=73
-36	105	-36+105=69
-42	90	-42+90=48
-45	84	-45+84=39
-54	70	-54+70=16
-60	63	-60+63=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-60\"$ and $\"63\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-60\"$ and $\"63\"$ both multiply to $\"-3780\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3t\"$ with $\"-60t%2B63t\"$ . Remember, $\"-60\"$ and $\"63\"$ add to $\"3\"$ . So this shows us that $\"-60t%2B63t=3t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"54t%5E2%2Bhighlight%28-60t%2B63t%29-70\"$ Replace the second term $\"3t\"$ with $\"-60t%2B63t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2854t%5E2-60t%29%2B%2863t-70%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6t%289t-10%29%2B%2863t-70%29\"$ Factor out the GCF $\"6t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6t%289t-10%29%2B7%289t-10%29\"$ Factor out $\"7\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%286t%2B7%29%289t-10%29\"$ Combine like terms. Or factor out the common term $\"9t-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"54t%5E2%2B3t-70\"$ factors to $\"%286t%2B7%29%289t-10%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"54t%5E2%2B3t-70=%286t%2B7%29%289t-10%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%286t%2B7%29%289t-10%29\"$ to get $\"54t%5E2%2B3t-70\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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