document.write( "Question 151358: factorize 5(3x+y)^2+6(3x+y)-8 please help me understand how to solve this sum. thankyou \n" ); document.write( "

Algebra.Com's Answer #111263 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let $\"z=3x%2By\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the expression goes from $\"5%283x%2By%29%5E2%2B6%283x%2By%29-8\"$ to $\"5z%5E2%2B6z-8\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5z%5E2%2B6z-8\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"6\"$ , and the last term is $\"-8\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-8\"$ to get $\"%285%29%28-8%29=-40\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-40\"$ (the previous product) and add to the second coefficient $\"6\"$ ?\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 $\"-40\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-40\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40\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 $\"-40\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-40)
\n" ); document.write( "2*(-20)
\n" ); document.write( "4*(-10)
\n" ); document.write( "5*(-8)
\n" ); document.write( "(-1)*(40)
\n" ); document.write( "(-2)*(20)
\n" ); document.write( "(-4)*(10)
\n" ); document.write( "(-5)*(8)\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 $\"6\"$ :\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	-40	1+(-40)=-39
2	-20	2+(-20)=-18
4	-10	4+(-10)=-6
5	-8	5+(-8)=-3
-1	40	-1+40=39
-2	20	-2+20=18
-4	10	-4+10=6
-5	8	-5+8=3

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"10\"$ add to $\"6\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"10\"$ both multiply to $\"-40\"$ and add to $\"6\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"6z\"$ with $\"-4z%2B10z\"$ . Remember, $\"-4\"$ and $\"10\"$ add to $\"6\"$ . So this shows us that $\"-4z%2B10z=6z\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5z%5E2%2Bhighlight%28-4z%2B10z%29-8\"$ Replace the second term $\"6z\"$ with $\"-4z%2B10z\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%285z%5E2-4z%29%2B%2810z-8%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%285z-4%29%2B%2810z-8%29\"$ Factor out the GCF $\"z\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"z%285z-4%29%2B2%285z-4%29\"$ Factor out $\"2\"$ 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( " $\"%28z%2B2%29%285z-4%29\"$ Combine like terms. Or factor out the common term $\"5z-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283x%2By%2B2%29%285%283x%2By%29-4%29\"$ Plug in $\"z=3x%2By\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283x%2By%2B2%29%2815x%2B5y-4%29\"$ Distribute\r
\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 $\"5%283x%2By%29%5E2%2B6%283x%2By%29-8\"$ factors to $\"%283x%2By%2B2%29%2815x%2B5y-4%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"5%283x%2By%29%5E2%2B6%283x%2By%29-8=%283x%2By%2B2%29%2815x%2B5y-4%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x%2By%2B2%29%2815x%2B5y-4%29\"$ to get $\"5%283x%2By%29%5E2%2B6%283x%2By%29-8\"$ .
\n" ); document.write( " \n" ); document.write( "

\n" );