document.write( "Question 473223: Factor the trinomial\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "6r^2 +29r -42
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #324512 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Looking at the expression $\"6r%5E2%2B29r-42\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"29\"$ , and the last term is $\"-42\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-42\"$ to get $\"%286%29%28-42%29=-252\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-252\"$ (the previous product) and add to the second coefficient $\"29\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-252\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-252\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-252\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-252) = -252
\n" ); document.write( "2*(-126) = -252
\n" ); document.write( "3*(-84) = -252
\n" ); document.write( "4*(-63) = -252
\n" ); document.write( "6*(-42) = -252
\n" ); document.write( "7*(-36) = -252
\n" ); document.write( "9*(-28) = -252
\n" ); document.write( "12*(-21) = -252
\n" ); document.write( "14*(-18) = -252
\n" ); document.write( "(-1)*(252) = -252
\n" ); document.write( "(-2)*(126) = -252
\n" ); document.write( "(-3)*(84) = -252
\n" ); document.write( "(-4)*(63) = -252
\n" ); document.write( "(-6)*(42) = -252
\n" ); document.write( "(-7)*(36) = -252
\n" ); document.write( "(-9)*(28) = -252
\n" ); document.write( "(-12)*(21) = -252
\n" ); document.write( "(-14)*(18) = -252\r
\n" ); document.write( "
\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 $\"29\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-252	1+(-252)=-251
2	-126	2+(-126)=-124
3	-84	3+(-84)=-81
4	-63	4+(-63)=-59
6	-42	6+(-42)=-36
7	-36	7+(-36)=-29
9	-28	9+(-28)=-19
12	-21	12+(-21)=-9
14	-18	14+(-18)=-4
-1	252	-1+252=251
-2	126	-2+126=124
-3	84	-3+84=81
-4	63	-4+63=59
-6	42	-6+42=36
-7	36	-7+36=29
-9	28	-9+28=19
-12	21	-12+21=9
-14	18	-14+18=4

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-7\"$ and $\"36\"$ add to $\"29\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-7\"$ and $\"36\"$ both multiply to $\"-252\"$ and add to $\"29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"29r\"$ with $\"-7r%2B36r\"$ . Remember, $\"-7\"$ and $\"36\"$ add to $\"29\"$ . So this shows us that $\"-7r%2B36r=29r\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6r%5E2%2Bhighlight%28-7r%2B36r%29-42\"$ Replace the second term $\"29r\"$ with $\"-7r%2B36r\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%286r%5E2-7r%29%2B%2836r-42%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"r%286r-7%29%2B%2836r-42%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"r%286r-7%29%2B6%286r-7%29\"$ Factor out $\"6\"$ 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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28r%2B6%29%286r-7%29\"$ Combine like terms. Or factor out the common term $\"6r-7\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"6r%5E2%2B29r-42\"$ factors to $\"%28r%2B6%29%286r-7%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"6r%5E2%2B29r-42=%28r%2B6%29%286r-7%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28r%2B6%29%286r-7%29\"$ to get $\"6r%5E2%2B29r-42\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );