document.write( "Question 216205: in class we are factoring trinomials. 7a^3b-35a^2b^2+42ab^3 \n" ); document.write( "

Algebra.Com's Answer #163272 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"7a%5E3b-35a%5E2b%5E2%2B42ab%5E3\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"7ab%28a%5E2-5ab%2B6b%5E2%29\"$ Factor out the GCF $\"7ab\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"a%5E2-5ab%2B6b%5E2\"$ \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( "Looking at the expression $\"a%5E2-5ab%2B6b%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last coefficient is $\"6\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"6\"$ to get $\"%281%29%286%29=6\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"6\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\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 $\"6\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"6\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6\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 $\"6\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*6 = 6
\n" ); document.write( "2*3 = 6
\n" ); document.write( "(-1)*(-6) = 6
\n" ); document.write( "(-2)*(-3) = 6\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 $\"-5\"$ :\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	6	1+6=7
2	3	2+3=5
-1	-6	-1+(-6)=-7
-2	-3	-2+(-3)=-5

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-3\"$ both multiply to $\"6\"$ and add to $\"-5\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5ab\"$ with $\"-2ab-3ab\"$ . Remember, $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ . So this shows us that $\"-2ab-3ab=-5ab\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%28-2ab-3ab%29%2B6b%5E2\"$ Replace the second term $\"-5ab\"$ with $\"-2ab-3ab\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2-2ab%29%2B%28-3ab%2B6b%5E2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%28a-2b%29%2B%28-3ab%2B6b%5E2%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%28a-2b%29-3b%28a-2b%29\"$ Factor out the GCF $\"-3b\"$ from the second group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28a-3b%29%28a-2b%29\"$ Combine like terms.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this means that $\"a%5E2-5ab%2B6b%5E2\"$ factors to $\"%28a-3b%29%28a-2b%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"7ab%28a%5E2-5ab%2B6b%5E2%29\"$ factors to $\"7ab%28a-3b%29%28a-2b%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "======================================================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"7a%5E3b-35a%5E2b%5E2%2B42ab%5E3\"$ completely factors to $\"7ab%28a-3b%29%28a-2b%29\"$ \n" ); document.write( "

\n" );